You misunderstand. Houston Methodist is not part of the government. No U.S. Federal or local government is going to coerce anyone to get the vaccine. Private organizations are free to coerce anyone they like. That's between them and their employees. By the same token, no one can force you to wear a mask, or put on a shirt and shoes, but a grocery store can ban you from entering without a mask, shirt and shoes. A grocery store is private property and they can make any rules they want, as long as they do not demand you do something illegal, and their rules are not discriminatory (such as banning minorities).
What about Covid passports? That is coming directly from the Whitehouse and the CDC