Is Keplers First Law a consequence of the conservation of angular momentum?

It would seem that once you have deduced that the angular momentum is conserved then you can deduce:

$r^2\dot{\theta}=h$ is constant

Combining this with the radial equation of motion then yeilds a differential equation whose solution is Kepler's First Law. So is Kepler's First Law a consequence of the conservation of angular momentum or am I missing something?

Well, yes and no. You need to use conservation of angular momentum but you also need to use the radial equation, which is specific for the case of gravity. A different radial force law would still conserve angular momentum but it wouldn't have elliptical orbits. Not to mention that conservation of angular momentum can be deduced from Newton's laws and the law of gravitation. So it doesn't seem very useful to say that the first law is a consequence of conservation of angular momentum, since you need a lot more than that to prove it. The same goes for the third law.

The second law, however, can be deduced just from conservation of angular momentum, so it holds for any central force, not just gravity.

No, not uniquely. Conservation of angular momentum is a necessary condition, but it is not sufficient.

Kepler's First Law says that the planets orbit in elliptical paths with the Sun at a focus of the ellipse. This specifically depends on

• an inverse-square law force and
• a negative total mechanical energy, with the reference zero for the potential energy is infinite separation distance.

This force and this energy are not dictated by conservation of angular momentum. Conservation of angular momentum results for any form of a central (aka, radial) force.

If the force is repulsive or the energy is too large, the orbit will not be elliptical. The first case would happen for like-signed charges orbiting each other (not planetary motion), and the second, a comet which executes a parabolic or hyperbolic orbit. These systems conserve angular momentum, but definitely don't follow Kepler's elliptical law., not