When someone is medically retired and unable to complete recruit training, can they still be considered a Marine?

No......just signing up, doesn't make you a Marine. You have to be able to pass all the requirements and be standing tall on graduation day before you can call yourself a Marine.

Not quoting any MCO here, but generally in the Corps, if you didn't make it through boot camp, you aren't a Marine. Regs technically will say you are, you're on the enlistment rolls, getting meals, lodging, pay, medical, etc & whatever separation paperwork you get sure comes from HQMC, but you didn't get through initiation. You may even rate lifetime VA bennies, but just use then and don't be trying to tell war stories at the Marine Corps League or VFW.
Just how it is.

LCpl Kenneth Bean Once you earn the Eagle, Globe, and Anchor, you are a Marine.