While attempting to refill a prescription, I was informed by message that the doctor would not refill without an office visit (which is normal), but that the office is closing permanently on Tuesday.

Aren't there issues with letting patients know that they'll have to get a new doctor? What becomes of medical records?

It was suggested to me that I look into state labor laws, which brought me here.

Thanks.

Medical providers are supposed to inform you when they are closing, give you a referral to at least one other professional, and let you know where your records are going.
Start looking on the website of your State medical board (may be called Board of Medical Examiners). They're the ones who monitor doctors' licenses and set the standards they must follow. It's not part of employment law.

Here's your state medical board:
http://www.tmb.state.tx.us/

Thank you very much.