Most of the post which I quote has been covered by others, but I wanted to specifically address
Also, as the velocity approaches the speed of light, the mass of the earth would increase, requiring a larger force to accelerate it. Assuming the force on the earth is constant, the acceleration would decrease, causing the effect of "gravity" to be reduced, and we would all float away.
Again, it's a question of reference frames. Just like velocity, acceleration, mass, energy, charge, etc.,
force is also relative. Force can be constant in our reference frame, but variable in another.
One way to look at this is to consider reaction mass (an explanation I will keep brief, since I am not an engineer). If you admit that the Earth's mass is increasing as its velocity increases, then you must also admit that, if the force that propels the Earth is rocket-like, the reaction mass (by which I mean the matter that is propelled "downwards" so that, by Newton's laws, the Earth must be propelled "upwads") should also increase in mass. Propulsion by a greater reaction mass at a given velocity produces greater thrust, which is balanced by the greater mass of the Earth.
The other way I like to look at it is to keep in mind that there ought to be no experiment that can determine my "absolute velocity". If you are correct -- in that our gravity should lessen as we accelerate, then I propose the following experiment: get in a rocket with no windows and accelerate upwards for a while with respect an inertial launch pad. After time ΔT, turn off the engines. You are now not accelerating and are therefore inertial. Since you are inertial, you are free to conclude that you are not moving at all, and that it is the launch pad which is moving away from you. Now turn the engines back on; you are effectively starting from zero velocity. Repeat this procedure as often as you like; in each instance, you should experience a force of pseudogravity. If you are correct, then after many instances, the pseudogravity ought to decrease because you have added up many accelerations and are getting closer to the speed of light. But this obviously violates the laws of special relativity: the "intermediate" inertial reference frames that you occassionally put yourself into by turning off your engine is "just as good" as the launch pad's reference frame. Thus we can conclude that we should be able to repeat this procedure indefinitely and never lose pseudogravity. In particular, we can let ΔT→0, and we essentially have the Flat Earth scenario (in which our "breaks" between accelerations have infinitesimal duration).
In sum: if you were right, relativity would be wrong.