I just replaced a cartridge in my faucet and when I tried to turn the water back on there is no running water even though I could open the valve under the sink. I tried putting in the old part to the same results.

Why did you replace the cartridge? Was there running water before? Back to you, Toim

Yes the faucet was dripping

It sounds like you disturbed some minerals built up in your pipe walls and now is clogging the inlet ports on your cartridge. Shut the water off and remove the cartridge. Check and clear the in let ports.
Now place a pan over the open valve body and turn the water back on for a moment to flush out the supplies. Good luck, Tom