Stefan Hartmann's letter to the Vatican, which he posted on his Facebook page, requested release from the traditional oath "in acknowledgement of my weaknesses and failures, with all due humility and after long consideration of my conscience and personal situation".
The 59-year-old, whose parish is in Oberhaid in north Bavaria, claims superiors recommended he rescind his own ordination after he revealed on a TV talkshow in January he had secretly fathered a daughter in 1989, eight years after vowing celibacy at Trier cathedral.
But Hartmann wants to remain and leave the oath behind.
"I still feel it is my calling to be a Catholic priest, a pastor and a theologian," he said in the letter.
In a foreword to the open letter, Hartmann said a positive response from Pope Francis, setting a precedent for Catholic priests to remain ordained outside of celibacy vows, would "bring solutions and relief in many cases”.
"There is a human right to partnership, marriage and parenthood, even if you can forego it willingly for religious reasons,” he wrote.
He added it could boost faltering numbers of young Catholics training for ordination, along with allowing back into the priesthood many ministers who had been forced to make a "painful exit" from the Church under similar circumstances to his.
Hartmann said he had been unable to stay in a relationship with the mother of his child and had kept the affair a secret to protect his career.
"After that I tried to walk the road of celibacy again, but since 2007 I have realized more and more that I am just not up to it," he said.
"I have known for some years now that the oath I took after just two and a half years of seminary training was too rushed, and did not reflect the constitution of my character."
Hartmann said, although he has no marriage plans as yet, the request was "above all to allow me to go into a marital partnership into which my daughter can be integrated as part of a family".
The Catholic leadership has as yet made no move to release priests from celibacy, but Vatican expert Andreas Englisch told Austrian paper the Kurier this month the question was among the changes Francis is most likely to make in coming years.
Pope Francis, who replaced Benedict XVI after the latter resigned in February 2013, has a reputation as a progressive church leader.