Could you get medically discharged for having high blood pressure problems?

Possible. That will be a decision made by your PCM and the Med Board if it gets to that. It'll certainly make you less likely to deploy to an austere environment.

It's possible to be discharged for "high BP" but usually one can get it under control with medication, dietary changes, and a good exercise regimen. Then if you maintain it in a normal level, they would have no reason to boot anyone.

It is possible but it all depends. Most SM's have a tendency to have high BP from time to time. If someone is suspected of high BP they will end up being put on a 7-14 BP screening and will get checked out. It will progress from there.