I don't believe there was a definitive resolution to Gamling's fate in either the movie or the book.
As you said, the last we see of him in the film, he's charging the Mumakil. He's not seen at the Black Gate, nor at the coronation (unless I missed him somewhere). His absence from those scenes might be taken as an indication that he was killed at the Pelennor, but we can't conclusively say that.
In the book, there's no mention of the character riding to Minas Tirith at all. He's last seen in the aftermath of the battle of Helm's Deep. It's possible that he was one of the Rohirrim who remained behind to protect Rohan while the rest of the army rode east.
The book also gives no definitive answer regarding his fate. One thing worth noting however is that there is a poem written after the war (song of the Mounds of Mundburg) which memorializes the fallen at the Pelennor and includes a roll-call of the more notable heroes who died there. Gamling's name is not among them. Given that the poem's author was from Rohan, I take that as a positive sign that Gamling did not die at the Pelennor (or, as hypothesized above, he may not have simply been there to begin with).
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