I'm curious why a starship should be compared to a conventional naval vessel for density. Is there some reason it needs to have a density of less than water?
Good point.
Personally, I prefer to compare starships to real-world spacecraft like the Soyuz or the Apollo CSM/LM. Been a long time since I did the work, but I figured out that the entire Apollo stack had a density of about 290kg/m^3, while the space shuttle was closer to 170 and Soyuz was around 210. Most newer spacecraft -- the Dragon, Cygnus, HTV and ATV cargo ships, for example -- have similar densities between about 150 and 250kg/m^3.
With a volume of 215,000m^3, that would give the Enterprise-A a mass of 32,250 to 52,500 tons.
If we're going to talk about common sense, the material descriptions we're given about starships is that they're designed to withstand weapon impacts that destroys relatively dense things like iron-nickel asteroids with ease. Even assuming that say torpedo technology evolved significantly between 2293 and 2370, an unshielded Enterprise-A was taking torpedo hits that would presumably destroyed much larger asteroids with relative ease. In which case the hull materials being less dense than a combination of nickel and iron doesn't make a lot of sense.
You're assuming that density is equivalent to physical resilience, which is not always (or even usually) the case. A starship with a 2-meter thick hull of deleted uranium isn't going to be more resistant to impact damage than a starship with 10 aluminum whipple shields 20cm apart. Add kevlar between the shields, you double its resistance; add trek-style forcefields between the shields, you make it virtually indestructible. Far more importantly, a 2-meter shell of depleted uranium is a lot harder to repair or replace or even service than a stack of thin wipple shields bolted to a frame; a photon torpedo may peel six of ten layers of your hull plating, but since it's just thin aluminum plate you can just unbolt the damaged sections and slap new ones in its place (whereas depleted uranium will crack/spall and possibly shatter as huge chunks of it are propelled into the inner hull at tremendous speeds).
Something to remember in space is that kinetic impacts are an action-reaction relationship. Putting more mass in front of a projectile simply creates more mass that can be thrown towards you when it hits. You want to DEFLECT the projectile, not stop it cold. Starships use deflector shields for this purpose, but their hull plating probably works in a similar way, in which case it is probably VERY low density and a lot more elastic than most of us would believe.
Indeed the requirements for a starship constructed in space (contra the absurdity of ST09) are probably more toward better thermal properties and maybe radiation shielding over things like lightness. There's simply no need for a starship that uses mass reduction technology to worry all that much about using high density materials for its hull.
Mass reduction costs energy. The less you have to reduce your mass to maneuver at impulse power, the more energy you have available for things like phasers, shields and sensors. As with real-world spacecraft: a heavier ship has a certain mass penalty as it burns through its fuel that much faster to get the same performance.
What's more, radiation shielding isn't accomplished (normally) with high density materials. Protection from radiation usually requires some material with a lot of hydrogen; LH2 and distilled water work exceptionally well, as does parafin or other forms of hydrocarbons. A ship with forcefield technology wouldn't have as much of a need.
There's nothing in canon or even official sources that suggest a 190,000 ton Enterprise, and there's evidence to suggest ships are, in general, heavier than water. There's no real reason for the compromises that naval vessels have to deal with in terms of hull density on a starship.
Naval vessels don't have to make much of a compromise, actually. They just have to displace more water than their own mass, which means there's a minimum size their hull components must be built to in order to achieve the needed displacement for their intended fittings.
Space ships start with completely different design assumptions. You start with mission capabilities, and then you go through a catalog to see what systems can provide those capabilities. Choose your payload, then choose a bus type (e.g. the physical body of the spacecraft) and then put it all together. The selection of the bus usually has a lot to do with what kind of propulsion system you have available, and a lighter bus gives you more propulsion options than a heavier one.
A lighter starship, by the same token, can get away with using a smaller impulse engine and a more efficient warp drive. Reducing the mass further might actually improve its combat maneuverability, and could also improve its thermal characteristics if the lighter materials absorb less heat.
