I don't see why it couldn't happen... but people who earn a medical degree usually use the medical degree to do some research to advance their careers. Once you become consultant level (meaning after you've done research to help you get to the top level of the medicine hierarchy), then you can really lead research. However, you wouldn't be making a career solely out of research...you'd be practicing clinically at the same time. So yes, you can make a career out of research but it is not separate from your clinical duties.
In the US, the same applies...you could do a research fellowship, and then once you reached attending level (equivalent to consultant) then you could lead research while practicing clinically. However, you cannot practice in the US without having been US-trained. That means, you must do a US residency and then deal with the immigration requirements. So the question of doing research is only linked to the MBBS in terms of immigration status, not degree status. An MBBS is equivalent to a US MD.
If you just wanted to go purely the academic route and not have any clinical involvement, then you'd probably be required to do as the PhDs do, with post-docs, etc etc etc.
