Maybe Florida bully (Sideroxylon reclinatum)?

Here is a link that might be useful: Bumelia (Sideroxylon) reclinata (reclinatum) Florida Bully

I saw something very similar today at the San Antonio botanical Garden and it was a type of myrtle. A friend of mine also has the this and she also said a myrtle but didn't know the variety.

Could it be cherry laurel? Prunus caroliniana

The fruit on the cherry laurel isn't the same I think.
Look at the stem part in a closer shot.
With the help of bostedo I believe it's one of these,

Chittamwood, Gum Bumelia, Woolybucket Bumelia, Gum Elastic, Wooly Buckthorn, Gum Woolybucket, Wooly Bumelia, False Buckthorn, Shittamwood, Ironwood, Coma
Bumelia lanuginosa

Gum bully (Sideroxylon lanuginosum) makes more sense than Florida bully (S. reclinatum). The linked Boerne NPSOT article discusses the variety of common names and how it frequently goes unnoticed in spite of being fairly common around the state.

Here is a link that might be useful: A Tree By Every Other Name Is Still The Same