Classically, earth's speed should increase continuously to create a force identical to gravity.

However, this is not necessary in special relativity. As force is the time derivative of momentum, actually "momentum" is what has to increase continuously. The relationship between speed and momentum is not linear at speeds closer to the speed of light, and is infinite at speed of light itself. So, our momentum can increase constantly in time, but our speed would never reach the speed of light.

Briefly calculating, our earth must be moving at a speed of approx. 298635.7263749823827 kilometers per second today, and still increasing.