Murphy admitted to the affair in September, and texts reported on by the Post-Gazette showed the woman accused Murphy of hypocrisy in January after an anti-abortion message was posted to his Facebook account.
"You have zero issue posting your pro-life stance all over the place when you had no issue asking me to abort our unborn child just last week when we thought that was one of the options," she wrote, according to the Post-Gazette.
A message from Murphy's phone to her in response said, "I get what you say about my March for life messages. I've never written them. Staff does them. I read them and winced. I told staff don't write any more. I will."
Asked about the report, a spokesperson for Murphy said, "The office has no comment or response to the story."
Murphy has touted his anti-abortion record over the years, and is a co-sponsor on a bill
that passed the House of Representatives on Tuesday that would ban abortion after 20 weeks.
Mary Lou Gartner, secretary of the anti-abortion group LifePac, said she considered Murphy, whom the group previously endorsed, an "honorable" person and cited the Bible when declining to criticize him in the wake of the report.
"I'm not ready to cast a stone at him," Gartner said.
She said the group's board would take these actions into consideration when the time comes to endorse for upcoming elections and would take into account how larger anti-abortion organizations respond.
Murphy had a 100% anti-abortion voting record from the Family Research Council -- a leading anti-abortion group -- based on the most recent congressional session it scored.
"Congressman Murphy's behavior is disappointing and disturbing," Family Research Council president Tony Perkins said in a statement. "His statements and actions are at odds with his own voting record. Inappropriate personal behavior has public ramifications -- and in this case, clearly opens him up to charges of hypocrisy. For his own sake and those around him, our counsel would be for him to bring his behavior in line with his past voting record, not change his policy stance to reflect his wrong behavior."
The Post-Gazette's report included excerpts of a six-page memo reportedly written by Susan Mosychuk, Murphy's chief of staff. The memo, written to Murphy and dated June 8, 2017, detailed a series of erratic actions and behavior that made Mosychuk and Murphy's staff fearful.
Describing one episode where Murphy berated the staff and threw documents to the ground, Mosychuk's memo said, "All the staff were scared of you."
The memo detailed another instance where Murphy read on his iPad and played YouTube videos as he drove in the rain.
"I will never get into a vehicle with you behind the wheel without an explicit agreement that you will not text or use your iPad," Mosychuk wrote.