After doing some research, it seems to be otherwise. The first 2
battalions of Marines came into being on November 10, 1775. They
were to serve as landing forces with the fleet. They were known as
the Continental Marines. However there was no mention to exactly
who the Marines were in service of. However, what is known is that
the term US Army did not exist at that time and technically neither
did the US Navy during the time it was created. IMO, since they
were the landing forces OF the fleet, that would mean they are part
of the fleet.
It is interesting to note that they do operate under Army Rank
though. They ranks such as "major" and not "commander'.
What really makes the Marines part of the navy are the traditions
they have ever since the founding. They use terminology that are
very "nautica". Terms such as "Aye." "Bulkheads" for walls,
"overheads" for ceilings, and "deck" for floors." Terminology which
are typically navy.
I do believe that the USMC began as Naval infantry.