The group, Catholics United, is targeting Roger Mahony, who last month was relieved of all church administrative and public duties for mishandling abuse claims against dozens of priests, dating back to the 1980s.
"Cardinal Mahony: Stay Home," the online petition reads.
"Your further implication in the church sex abuse scandal and being barred from public ministry in the Archdiocese of Los Angeles should be an indication to you that you should not attend the next Papal Conclave."
"If a Cardinal is stripped of public ministry in his diocese, why should he be rewarded with being allowed to vote for the next pope?" the group asked.
"Cardinal Mahony would further increase the scandal and shame in our Church by attending the Papal Conclave."
According to Catholics United, Archbishop Gomez is in favour of Cardinal Mahony's attending the conclave that will choose a successor to Pope Benedict XVI, who resigned unexpectedly last week.
Cardinal Mahony, who is now 76, took part in the vote that chose Joseph Ratzinger as pontiff in 2005. He took the name Benedict XVI.
And Cardinal Mahony is theoretically still a member of the conclave made up of 117 cardinals aged under 80 who will meet behind closed doors in the Sistine Chapel until a new pope is chosen.
This month the Los Angeles archdiocese released files on more than 100 clerics accused of sex abuse, as required under a 2007 lawsuit deal, after records were published showing Cardinal Mahony discussing how to cover up alleged crimes.