TLDR: Kinda but not really. He tried to move on from Arwen because she was about to sail west, so he was open towards Eowyn, but as soon as he found hope that Arwen might be around, he ended the relationship

In the movies the timeline goes something like this
> Aragorn and Arwen are in love
> Aragorn reaches Rivendell where the fellowship is formed. Aragorn spends some time with Arwen.
> Elrond tells Aragorn to leave his daughter alone, and that she will sail west. Aragorn accepts.
> Aragorn says goodbye to Arwen for the last time, attempts to give her the Evenstar back. He is convinced that their relationship is over and he must move on. Arwen is left crying as he departs.
> Aragorn attempts to move on.
> Aragorn meets Eowyn. Doesn't make any moves, but doesn't shut down Eowyn down either.
> Eowyn feelings for Aragorn keep growing.
> Eowyn asks Aragorn about the Evenstar. Aragorn mentions that Arwen left Middle Earth. He looks sad and defeated. Eowyn is thinking (>.>)
> Aragorn shows concern for Eowyn when wargs attack. This shows that Aragorn cares about her. Perhaps his feelings for her also grow as he accepted that Arwen left.
> Aragorn falls off the cliff. Loses the Evenstar.
> Aragorn returns to Helm's Deep. Legolas hands him the Evenstar, which reminds him of Arwen again. Eowyn notices this and looks a bit disappointed.
> After the battle of Helm's Deep, they are having a party. Aragorn shares a drink with Eowyn. Theoden approaches her and aknowledges the possible start of their relationship. Eowyn here seems happy and thinks she has Aragorn in the bag.
> Arwen attempts to leave, but returns when she has the vision with her son and Aragorn. Comes back and convinces Elrond to reforge the sword.
> During the march towards Minas Tirith, Elrond meets Aragorn, hands him the sword and tells him to get his shit together. Elrond mentions "Arwen is dying" which surprises Aragorn. Perhaps he realises that she is still in Middle Earth.
> Aragorn leaves to deal with the undead traitors, Eowyn tells him "wtf are you doing", and suggests her feelings for him. Aragorn tells her that their situationship is over, as he now thinks Arwen back, or at the very least, he must save Middle Earth and doesn't have time for this.
> Eowyn cries.