The Marines being under the Department of the Navy is largely irrelevant in considering this question. Requesting an interservice transfer into the Navy would be the first step a Marine who wants to be a SEAL would have to take toward achieving this end. So the answer to the question is technically yes, but bear in mind that someone in the Army or Air Force can also pursue the exact same opportunity.
In fact a significant number of people I served with while enlisted in the Marines had gone on to request and receive interservice transfers into the Army with the intent of trying out for special forces teams therein like the Green Berets. I guess once their efforts at getting into Force Recon were exhausted they figured they might have an easier time of some other special forces assignment as the Army provides. A couple of them made it, but the rest were then stuck on the Army side of things (because there’s no backsies with an interservice transfer) boring their fellow soldiers to death with stories about their glory days when they were in the USMC*.
Otherwise, it should be understood that interservice transfers – regardless of whether special forces are involved – should not be looked at in the same way as a promotion in rank. A Marine who wants to become a SEAL has to initiate the steps himself to make this happen, and then work insanely hard to see it through, but it will never be the case that a Marine (or Airman, remember, for that matter) suddenly finds himself “promoted” onto a SEAL team the same way a baseball player from a farm team might be called up to the major leagues.
*Before I hear it in comments – Yes, I know: “Once a Marine, always a Marine.” and I agree, but I further personally believe there’s some value to getting to say “Once a Marine, always a Marine, and only a Marine”.