Can I use rogaine to regrow hair that I ripped out around the nape of my neck?

You can use Rogaine shampoo for hair growth and if there is high damage on the area, than you can  apply Minoxidil to regrow hair on that area.

Rogaine  will work if there are follicular units left in your scalp. If you have ripped out the follicular units it will not be regrow. Since the side effects from Rogaine are minimal, you can try to use the 5% foam for a  period of 12 months. Do not stop prior to this duration.PRP hair MD.com

Rogaine will only work if you still have follicles left in the area to stimulate. If they are not present, no amount of drug will bring it back.  You may want to consider having a consult with a hair restoration physician who will look under a microscope to see if there is at least miniaturized hairs in the area.  Perhaps there is inflammation, scarring or traction alopecia ....it is hard to say without an in-person hair/scalp exam as well as a detailed history.

Assuming the hair follicles are intact, you can certainly apply topical minoxidil (Rogaine) 5% foam to the area to help stimulate hair regrowth. If the hair follicles are no longer present due to ripping them out (a.k.a. traction alopecia), then topical minoxidil will not help with regrowth. You can certainly give topical minoxidil a 6-month trial because it has a low side effect profile. Use it everyday and apply it to a dry scalp in the area where hair growth is desired. 

The use of minoxidil may work on the nape of your neck. You can try it but stick to it for at least 6 months