It is a Live Oak.
The bark on the stump looks like so many other oaks. The leaf size is correct too.
Brazilian pepper is at most an overgrown, dense shrub. In my 9 years in Florida and being exposed to thousands of peppers, not one ever came close to the girth that you see from the cut wood.
The image of the tree is classic live oak with dark green, rather small leaves. As they get really big, their foliage at times appears gray green or silvery green in color.
