Can a involved father who was there when he was born sue mother for not putting his name on birth certificate or his last name? We live in California.

You can sue anyone for anything, whether you have an acceptable cause of action or could win is another matter entirely.

Has paternity been positively confirmed ?

No, since it is up to the mother how to fill out the birth certificate. What you CAN do is file for paternity to be legally established and for joint custody.

Unless you were married at the time of the birth or so many days before the birth, the mother is not required to give the baby the supposed father's last name or have him named on the birth certificate.

The mother in most areas are completely in control of the birth certificate. As the father, you can file to prove paternity in court and ask court to add you to the birth certificate. The court will rule you are the father if you pass the paternity test at that point you can arrange support and visitation agreements

The mother in most areas are completely in control of the birth certificate

As long as the mother and father were not married. ;)