stingc

No, the earth would rip itself apart long before reaching the speed of light.

First of all, think of the scalings involved. The mass would not increase enough to keep things together. The surface will become weightless at a speed of 3.1 km/s, which is much less than the speed of light. That assumes that the earth is increasing radius without adding more material.

If the earth kept the same density as you increased its size, then the orbital velocity problem never comes up. Instead, it would collapse into a black hole long before reaching 4x10^12 meters in radius.

Next, you're applying a concept which has no validity here. Special relativity does not apply to gravitational problems, and you have to be extremely careful in applying it to anything involving acceleration at all. Analyzing exactly what would happen requires general relativity, and is an extremely difficult problem.