If you correct an iodine deficiency will hair grow back?

This really depends on the specifics of your situation, like how long you were deficient, how long ago you lost your hair, and how severe the deficiency was. It will definitely stop further loss and likely regrow some of the hair but it probably won't grow all of it back. But, to know for sure you need to talk to your doctor (or just wait it out and see).

If your hair loss is due to iodine deficiency, correcting the deficiency should at the least prevent further loss but may not result in the lost hair returning.

This is a more complicated question than it might appear at first. First, I'm always interested when one presents with iodine deficiency how it was actually diagnosed (urinary iodine testing vs other) and whether there is clinical or biochemical evidence of thyroid  dysfunction. In the absence of thyroid abnormaties, many patients find their hair does not improve fully with routine iodine supplementation. The question however has many variables so please see a knowledgeable hair specialist.

Correcting the iodine levels may or may not reverse the hair loss. It will take at least 8 months to find out the answer to this question