If you read rhe comic you'd know it took several issues of him trying to stop her, grounding her, etc... But Peter realizing she was as stubborn as her at her age and she was close to being an adult is partially why he accepted what she did.
Also Mayday's decision to be Spider-girl stems from the fact that she stopped a villain who woukd've killed Peter and in stopping that villain she saved his life. It's an inverse of the Spider-Man origin. Peter learned that if he didn't do anything people died, Mayday learned that if she did something people lived.
It's also why Dan Slott is a hack.