Can my leadership request my personal medical records?

Being in the military does not exempt your command from federal HIPAA requirements. If the Army owned your records as other suggested, then your command has only to request the tests from your provider. There are certain times when this happens, and those specific records will be released to the command. For instance, if you're caught doing drugs on a lab test. Anything that your command is authorized to have, they don't need your permission to access.
You are well within your rights to tell your command they can request anything they need from your provider

You're lacking the context of why they're asking. Are you missing work, do you need surgery, are you limited because of XX condition, what kind of tests are they trying to get info on? Generally no, you don't have to give them all that personal information, that's why PAs issue physical profiles. However, there are times especially if they suspect drugs or alcohol in reference to a situation etc where it would be appropriate, but that usually requires a commander to inquire.

I think the short version here is: If your command needs your records, and is authorized to receive you records, then they can ask the hospital for those records and get them. Having YOU do it and bring it to them doesn't sound quite right. I'd request that they ask for them from the hospital directly.