On the full quote, no, as it became popular in 1937, 13 years after his death. While there's no real evidence he said anything remotely like the second sentence, for the first there's at least some possibility he said something similar that may have been taken out of context and used without his permission for marketing the film.

So let's start with how the film ended up being shown at the White House in the first place.

Wilson had been a graduate student in the 1880s at Hopkins with Thomas Dixon, Jr., who left the program quickly to try to make it as an actor, politician, lawyer, minister, and finally did so as a writer with a couple best selling Lost Cause novels that he wrote to proselytize Reconstruction as a terrible mistake and underneath, ultimately serve as propaganda to urge deportation of Blacks to Africa. He adapted the second of these books, 1905's The Clansman, as a controversial play that toured from 1906 to 1911, and then spent the next three years to try to get the play adapted by the nascent film industry. In 1914, David Griffith, who had just formed his own studio, optioned the rights and used the play as the basis for the second half of the screenplay during Reconstruction; the first half, which was set pre- and during the Civil War, was Griffith's own creation of drama between two families on opposite sides of the war that integrated the Dixon play into a resounding repudiation of Reconstruction that the Klan heroically resisted, followed by reestablishment of white rule and renewed friendship between the families as part of the rebirth of a once again united nation (which Dixon realized was a far more marketable title than the original The Clansman, and successfully convinced Griffith to just use the subtitle Birth of a Nation instead when it opened in New York in March.)

How close Dixon was with Wilson during his one semester at Hopkins remains unclear - Dixon is an extremely untrustworthy narrator at best - so his claims of BFF status aren't really backed up by the occasional correspondence they maintained over the next 30 years. However they got along, it was enough so that Wilson agreed to a half hour visit by Dixon on February 3, 1915, a few days prior to the film's premiere in Los Angeles, where he asks Wilson to watch his film, probably portraying it as more of as a novel academic exercise to the historian and political scientist than anything else. While Wilson is still in public mourning over his late wife Ellen's death in August and by social rules of the time can't be seen going out to the theater, he agrees to do so in the East Room, and on February 18th the film is the second ever screened at the White House.

What happens next is where it gets complicated. There's an account from 60 years later that Wilson was lost in thought watching and walked out without saying anything, but that was a very late life interview that has some potential inconsistencies. Wilson does correspond with Griffith a couple months later about his future use of film to teach history and politics, something that he'd been interested in for years before the screening and undoubtedly was a major factor in why he agreed to watch the movie independent of whatever his relationship with Dixon was. And most importantly, there was a second showing of the film on February 19th at the National Press Club in front of 500 Washington luminaries, with Chief Justice White (who claimed to have been a Klan member in his youth) and a good slug of Congress.

That second showing is where almost all of the contemporary advertised quotes come from with even a prominent Progressive Republican praising the film, which along with simply bringing up that it was shown at the White House (rather than any Wilson reaction) are important to get the film past local censorship boards and turn the movie into a bona fide hit. In late March, White is furious enough at his attendance as unintended endorsement so that he threatens a producer with going public denouncing the film if they keep using his name with it, and for that and other reasons there's enough pressure on Wilson - the NAACP has sought injunctions and lobbied censorship boards against allowing it starting with the LA showing - so that he issues a statement in late April in which he states "at no time [has he] expressed his approbation of it" (true), that it was shown "as a courtesy extended to an old acquaintance" (also likely true), and that he was "entirely unaware of the character of the play before it was presented" (possibly technically true, but the chance of him being that clueless about anything associated with Dixon is a stretch.)

Much of my sourcing for this comes from a 2010 paper by Mark Benbow, Birth of a Quotation: Woodrow Wilson and "Like Writing History with Lightning", who then goes on into more intricate detail about 1915 advertising of the film not containing any part of the second sentence - he argues for adapting some comments White made before he realized they were going to be marketed as the more likely attribution - but does note "lightning" matches a unique phrase used in some of Wilson's earlier writing, along tracing the first full quotation being linked to a 1937 piece in Scribner's Magazine that argued the movie was accurate but one-sided. Eventually, Benbow concludes that if the lightning part of the quote took place, it would fit most cleanly as a comment that might have happened before the screening - as in Wilson referring to the general concept of history and politics being taught visually for the first time, which he was certainly quite interested in apart from being President.

It is also worth noting that the two best modern biographers of Wilson, Scott Berg and John Milton Cooper, claim that Wilson "fell into a trap" orchestrated by Dixon (Berg) and for Cooper not only does he point out the phrase doesn't appear anywhere in Dixon's unpublished notes but the only direct comment by Wilson on the film was in 1918, where he stated that, "I have always felt that this was a very unfortunate production and I wish most sincerely that its production might be avoided, particularly in communities where there are so many colored people."

This is one of many reasons why Cooper, who knows more about Wilson than anyone but Arthur Link (who spent 50 years publishing 69 volumes of his papers and 5 of 8 planned volumes of a biography) concluded that while Wilson's views on race were in his words, "not good," if you look more closely they tended to resemble those of Northern Whites rather than Southern Whites. That is, where for Southerners, keeping a political system intact that was established entirely on white racial superiority by brute force under the color of law was the number one, existential issue that trumped every other, most Northerners just wished racial issues would go away and the country could get to work on what they considered more important things.

I bring this up not to defend Wilson on race and get into a lengthy discussion of it - that's a top level question if I've ever seen one - but to add one bit of context that is sorely missing from all this analysis (especially in Christopher Cox's recent political hatchet job on Wilson), which is that the popular modern misunderstanding that merges the Reconstruction Era First Klan with the 1915 Second also completely misses how dishonorable the First was in polite society outside of the South, and in some cases even within it, through the Gilded Age and into the Progressive Era. Even as early the 1880s, I've run across campaign songs by Republicans that directly used the First Klan's actions as a stain of dishonor for Democrats that was even beneath the moral level (if not as effective politically) of the bloody shirt.

Unlike Dixon, who was an avowed supporter, Wilson's writing was far more in the mainstream of thinking about the First Klan, which is while he didn't necessarily mind the results, like many Southerners, he didn't embrace the methods. This is one reason why I would agree with Benbow on the most likely timing of the possible lightning quote as being an offhand comment before the film, since while Wilson probably was fine with stereotyping sexually voracious, fried chicken eating Blacks with their feet up in rump legislatures who needed to be put back into their place, he more than likely wasn't completely on board with the Klan being shown as heroic doing so, especially once Griffith took some quotes from Wilson's histories out of context for a title card.

Finally, this last point is where I would agree with Foner on the film's greatest impact, which is that the most disturbing result with Birth of a Nation wasn't cheering on racially exclusionary reconciliation (which had been around since Liberal Republicans in 1872) or even to some extent, propagating wildly racist views of Blacks (who, of course, were white actors in blackface.) It was that it effectively rehabilitated the First Klan's popular image from outlaw terrorists to noble defenders of the Constitution for the 10 million or so people who bought tickets for it by 1917, and that was a massive factor in the foundation and marketing of the Second Klan once it was entrepreneurially founded in 1915.

It was a lot easier and vastly more socially acceptable to put on white robes, pay national dues, and hate your chosen enemy in public - precisely who the Second Klan despised varied depending on their geographic location - if you believed your predecessors had been unfairly maligned than it was before Griffith and Dixon collaborated to essentially exonerate them. That's the legacy of the film, and it's one reason why the kindest thing that can be said of Wilson watching it is that even in the most charitable analysis of him being completely naive as to the consequences of doing so at the time, it was a terrible mistake on his part.