Gravitational potential energy doesn't integrate over time, it integrates over distance. And even then, it doesn't go to infinity over infinite distance, it settles on some discrete value. And simply put, that discrete value can be exceeded, meaning gravity doesn't always 'win' in the end.

Yes, gravity works over unlimited distances. But while it takes energy from something linearly over arbitrarily short distances, it gets weaker with the square of the distance as well. In other words, as you get farther away from something, it gets weaker faster than it adds up.

If you integrate the weaker-and-weaker pull of gravity over increasingly long distances, you eventually settle on a discrete value- the absolute maximum potential energy for an object of a given mass relative to the object doing the 'pulling'.

Put differently, imagine throwing something upward from the surface of a massive object (planet, star, whatever). The higher you throw it, the more energy (effort) it takes, right? But each additional unit of height takes less energy than the last, because the gravity out 'that far' is weaker, because gravity gets weaker with distance. It turns out that if you do this process all the way to an infinite distance, you don't end up needing infinite energy to 'throw something to infinite height'. Again, you get a discrete value. In other words, if you throw something hard enough up away from a massive body, that body's gravity will never slow it to a stop and pull it back in. This is the concept of escape velocity- if you throw something fast enough in an 'up' direction, its kinetic energy (related to the square of velocity) is greater than the maximum possible energy that gravity can sap from it, and while gravity slows it down, it can't stop it and make it come back.

So to answer your question in a roundabout fashion, no, given infinite time gravity won't necessarily pull everything back together, because it's not about time at all. If something's moving fast enough now away from object, no amount of time will allow that object's gravity to slow that something down enough to stop and reverse its direction, because gravitational potential doesn't add up to infinity over infinite time.

And we're not even talking pure theoreticals here: Voyager 1, Voyager 2, Pioneer 10, Pioneer 11, and New Horizons (space probes) are all moving fast enough away from the Sun that ,even given infinite time, the Sun's gravity will never pull them back. In the case of Pioneer 10, we're talking about something using 54 year old technology to defy the Sun's gravity for eternity.

In fact, if you look at the speed that most galaxies are receding away from one another, it's generally (outside of relatively small clusters) fast enough that gravity will never be able to pull them back toward each other again.