No, because we’re not part of the Navy, we’re part of the Department of the Navy, there’s a subtle difference, & not too many people know the contradistinction. Some people will say they’re the same thing, & I say no, they’re not, for one important reason. When I hear someone say that as a jibe to the Marines, I respond with, “Who is in charge of the Navy?” and they being in the Navy will usually respond with, the CNO or Chief of Naval Operations, who is an Admiral or 4-star flag officer. Here is the Seal of the Navy.
Then I ask them, who is in charge of the Marine Corps, the correct answer being the CMC or Commandant of the Marine Corps who is a General or 4-star flag officer, same rank as the Admiral who is the CNO, which means he is of commensurate rank. Here is the Seal of the Marine Corps.
Now here’s where it becomes interesting because then the next question I ask them is “Who is in charge of the Department of the Navy?” to which they usually don’t have the answer, so I’ll reply with the SecNav or Secretary of the Navy which is a Cabinet-Level Position within the president’s administration just like the Secretary of the Army or Secretary of the Air Force. But if they do have the correct answer, I’ll reply with, “Are the same person, or to be more clear, is the CNO the same person as the SecNav, which no they are not, because one is a military position held by a ranking officer, and the other is a civilian cabinet-level position held by an appointee. Here is the seal of the Department of the Navy.
Notice they are all different.
So while the Navy is part of the Department of the Navy, and the Marine Corps is part of the Department of the Navy, they are not the same thing, think of them like two fingers on the same hand. Both the CNO & the CMC report to the SecNav, the CMC does not report to the CNO so the chain of command is clear.
Telling the last part of this I use the forefinger for the Marine Corps & the middle finger for the Navy for the end of the joke which is FTN or F The Navy, which is my way of getting back at squids who used to like to needle us jarheads with that question.
One more little note, purely my observation and I’m just throwing this out there. If your a God fearing man or woman, you’ve most likely heard of the Holy Trinity.
The Christian doctrine of the Trinity (from Latin trinitas “triad”, from trinus “threefold”)[1] defines God as three consubstantial persons,[2] expressions, or hypostases:[3] the Father, the Son (Jesus Christ), and the Holy Spirit; “one God in three persons”. The three persons are distinct, yet are one “substance, essence or nature”.[4] In this context, a “nature” is what one is, while a “person” is who one is.[5][6][7]
(Thanks Wikipedia)
Now go look at the logos up top on the Department of the Navy web site. OOH RAH!!!!
As The Rooster Crows 
That is a very detailed answer. My head is still spinning. Surely no one would dare argue with you about it given your clear knowledge of the issue. (You even managed to sneak the Trinity into your explanation at the end.) 🙂
Equal footing under the law Bunk. lol