The forces generated in tennis from stopping, starting and changing direction means only the brace will provide meaningful ankle protection.
Some players find an ankle brace can allow them to return to play without the likelihood of soon suffering another ankle strain.
Are you playing on a school team? Tennis players sometimes don't realize there are often physical therapists that can check out problems like ankle sprains, and offer reasonable advice and therapy.
Indeed, it is always better to be seen locally and actually examined by a therapist or sports medicine specialist than to just "wing it" by posting, and trying to figure out how bad your injury is, how best to recover, when to return to play, and what type of strengthening you can do all on your own.
As ollinger points out, one of the potential problems with an ankle brace is t that is the ankle is fixed, so that the stress can be transferred to the knee.
After you ankle no longer has pain, you can do "ankle strengthening exercises" (the ankle is actually a joint and as such has no muscle, but strengthening the leg muscles can take stress off the tendons and ligaments, and over the time the ligaments themselves can tolerate greater forces):
Have You Suffered A Sprained Ankle?
http://www.joint-pain-solutions.com/images/anklestrengthening.jpg
Eventually, agility drills are a great way to develop "strong ankles", as well as to improve your on court movement:
USTA agility drills:
http://assets.usta.com/assets/1/USTA_Import/USTA/dps/doc_437_269.pdf