can we grow a new system?

Your first sentence is wrong, actually. We can't do that with current technology.

For one thing, there aren't a central "pool" of stem cells which produce all the cells in the body. There are distributed stem cells in all of your tissues which replace cells as needed. I suspect that replacing "all" the cells in your body within 3-6 months might be a little optimistic, too - certainly for fast growing tissues like your intestines or your skin, that's probably true, but it's likely to be much slower for longer lived cells. There's also a question of what you mean by "replaced" - clearly, the stem cells won't be replaced (although they will have divided a few times). In order to do what you have in mind, you'd need to target each individual stem cell niche at the same time and not only add in new ones, but kill off the old ones. I'm really not sure how you'd go about doing that.

What would be the consequences? The major issue is with the immune system. The immune system is primed to recognise "self" cells from "non-self" cells. So if you did that all at once, one of two things would happen - if your old cells predominated, your immune system would wipe out the new cells, leaving you with no stem cells and you would die. If the new cells predominated, you would get graft-versus-host disease (which is a real thing - the only time we do stem cell transplants for routinely at the moment is with the immune system, which will obviously recognise YOUR cells as "non-self" and start attacking) - and you would die.

But lets imagine for a moment that you somehow survived. Would it produce a new reproductive system? No, 'fraid not. The macroscopic shape of tissues and organs develops in the uterus, there's no way for your body to generate that shape again from new. If you were adding stem cells in, those stem cells in adults are designed to regenerate tissues that have already formed, not produce entirely new ones.

I will say that I'm not immediately sure whether there would be any difference at all in your sexual organs. Men and women actually have all of the same genes, and it's only very slight nudges at a very early age that start the ball rolling towards one gender or the other; once you have hormones in the mix, the hormones that are there tend to keep things running more or less as they are. I suspect that your cells would behave exactly like they would otherwise. Worst case scenario is that they would just stop responding to your sex hormones, and you'd go sterile. You certainly wouldn't magically grow a vagina.