Starfleet does have a lot in common with the navy, but it is not purely a military force, but also humanitarian, exploratory, scientific and diplomatic entity as well, so a certain degree of familiarity would be expected.
Of course it would entirely depend on who was in command. There would be some CO's who would adhere to strict protocols and others who would be more relaxed in how they address their crew and allow themselves to be addressed.
And as The Mirrorball Man
said, why does Starfleet have to conform to exact way of the US navy? You have the traditions and cultures of other planets to bring together when the Federation Starfleet was being created.