Yes, that was one of Roddenberry's claims. It makes more sense than the other reference that he used ("Wagon Train to the stars").
There's a bit of old seafaring language sprinkled into the Star Trek franchise, though some of it is obscure and re-tasked.
For example, warping a ship (usually *out* from a quay) means moving it by hauling on a line that is fastened to or around a piling, anchor, or pier.
BTW, if you've seen the recent film Master and Commander: The Far Side of the World, you'll notice a continuation of the captain / ship's doctor friendship as a plot element. I can't recall if that was present in Hornblower; I haven't read that since I was a child.
If I were on a ship and at risk of injury or death I’d want to be friends with the doctor too. Doctors are people and like most people will try harder to save someone they are friends with.
IIRC, "warp" drive consists of somehow attaching the ship to a (moving or generated) gravity wave. So the idea of pulling the ship along does, in fact, properly use the term "warp."