In usual flat space QFT, you don't have gravity at all, so no!
Well, QFT can be also be safely done on a curved spacetime background, but you are right, you don't get dynamic gravitational effects from it. What I implicitly assumed is QFT+semiclassical GR, where one uses semiclassical probability-weighted stress-energy tensor as a source.
This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous thread is at close to 500 comments.