If the Moon had gravity as strong as the Earth's, and a magnetic field, could it have supported life? If the Moon had gravity as strong as Earth's, and a magnetic field, could it have supported life? Because if the Moon had as much gravity as Earth, it could have retained more water than is present today on the surface.
If the Earth is in the habitable zone, does the Moon also lie in the habitable zone?
 A: If the Moon were exactly the same as the Earth, then sure, there is no major reason to suspect it would be any different. It is in the same orbit around the Sun as us, so it gets heated by the same amount. This would place it in the habitable zone.
However, habitability is not the same as being in the habitable zone, and the detailed answer depends on how you make the surface gravity match that of Earth. The surface gravity of a sphere of radius $R$ and average density $\rho$ is
$$ g = \frac{4\pi}{3} G \rho R. $$
Most rocky bodies in the Solar system have about the same density - that of a rock - so making the Moon's gravity match the Earth's is just a matter of making it bigger. Essentially it would become Earth's twin in every way.
On the other hand, maybe you intended to keep the size the same. In that case you would have to increase the density. It is not clear what you would make the interior out of, but it is pretty certain you will not get the same geology as on Earth. For one, smaller bodies cool off too fast to be geologically active at this age (roughly 5 billion years). You see, when the planets condensed out of the gas and dust swirling around the Sun billions of years ago, they were hot - gravitational potential energy went down, and so thermal energy went up. Their heat capacity is proportional to their volume, but their heat losses are proportional to their surface areas. Thus objects with high surface area-to-volume ratios (i.e. small things) cool quickly. The thing is, Earth's geologic activity probably had a large role in building up and maintaining the atmosphere and oceans we know and love.
In either case, there is also the problem of tidal locking. It is suspected by some that having tides was crucial for the development of life. The Moon is already tidally locked with the Earth - we only ever see one face of it - so it has no tides. If you scaled it up, you might tidally lock the Earth as well. The Moon would essentially be in a geostationary orbit, and we would not have tides. This is the case for the Pluto-Charon system, for instance.
