Well, if gravity suddenly stopped, yes, but if we had no gravity at all the Earth probably wouldn't be going around the sun anyway and we'd all be dead/not living.

There wouldn't be any sun to begin with either. The sun is basically a huge H-bomb kept in check by its own gravity.

Your tangential speed varies with latitude, ranging from 1,000mph at the equator to zero at either of the poles, where you wouldn't be flung off at all.