Could be, but down south more likely sugarberry. Common hackberry fruit usually is a dull purplish-maroon color when ripe. Pic below is sugarberry.

Anywhere in the SE:

Could be Celtis tenuifolia.

Anywhere in Texas & South & East:

Celtis laevigata

Anywhere in Texas & South & SW:

Celtis reticulata

Anywhere in Texas:

Celtis lindheimeri

It wouldn't surprise me if Celtis occidentalis would grow down there but I'm too lazy to do a lot of comparing of these plants right now.

Check them out. ;-)

I don't think that is a Celtis sp. at all. Compare with Ilex decidua - Possumhaw Holly.

Were you able to collect fruit? The seed(s) inside would also tell the story.

Great point VV.

I was reminded when I grew some Celtis occidentalis from seed, the seed looked like little, walnuts inside the seed coat. Similar to what these celtis laevigata seeds look like below.

OTOH: Here's a picture of Ilex decidua seed, a little more banana shaped and more than one seed inside the seed coat.

{:-)

ty