The 30-year-old was spotted by several witnesses urinating on the car after he left premises in Byron Bay, on Sunday.
On Monday morning, local police confirmed that he was issued with a criminal infringement notice.
“It put a dampener on our wedding weekend. I am incredibly embarrassed and disappointed in myself,” Bird told reporters.
“After getting married on Saturday night down in Byron Bay, my wife and I caught up with friends and family at the Beach Hotel on the Sunday night for dinner, after which we went to our car parked down near the beach and actually adjacent to a police car.
“I stupidly and regretfully went to the toilet in between the two cars, in the space and obviously facing away from my family.
“I know it was a stupid and dumb thing to do but I really want to emphasise the fact that I didn’t intend to disrespect anybody and I definitely didn’t do it on the car.
“I would like to apologise to the club and its fans, anyone I may have offended, especially to my wife Becky.”
Titans chief executive Graham Annesley admitted that the incident was “embarrassing” and “unacceptable to the club”.
He also confirmed that Bird would be subject to club disciplinary procedures.
“We’ll continue to talk to the NRL and we’ll hold a club disciplinary meeting later this week so we follow due process,” Annesley said.
“It is to Greg’s credit that he has publicly apologised and clarified exactly what took place. That will all be taken into consideration before a decision is being made.”