Petyr is known to have challenged Brandon Stark for the hand of Catelyn Stark. He was a much small and a callow boy when compared with Brandon Stark who was much renowned for his swordsmanship in the seven kingdoms. Hence it was no surprise when Brandon defeated Petyr. But suppose Catelyn had wanted to marry Littlefinger and chose to lope with him, it would have resulted in a shame for the House Tully. For the House Tully always stood to honor their words and the women of that house were not allowed to chose who they were to marry. Petyr Baelish was a ward of Hoster Tully and grew up with the Tully children, falling in love with Catelyn through the years. Had he been able to persuade Catelyn to marry him, there would have been a very different circumstance where neither of the Stark children, Robb, Sansa, Barn, Arya or Rickon, would have been born. Ned would have been married to some other Lady of a house. He would still have Jon Snow as his bastard, because of his promise to Lyanna Stark. Petyr may not have been so much ambitious as he is now in the series. Catelyn would have still been alive in the show if she had married Littlefinger as he would ahve done anything to keep her safe.
2. Petyr loved Catelyn and she was the only woman he ever loved. Everything he ever did, he did for Cateyn. He even risked his own life when he challenged Brandon Stark to a duel fo her hand. He would still have been the Master of coin and haeld a place in court. Catelyn never would have gone North. She would be living at King’s Landing with Littlefinger. Since her sister Lysa had married Jon Arryn, she would have taken an interest in the death of Jon Arryn.
Ned would have also come down south as the new Hand of the King. Knowing that the new Hand is a stranger in the South and a brother to the one whom she was betrothed to, Catelyn would have helped Ned. This would have angered Petyr as he had always hated the Stark brothers. So he would have obviously helped the Lannisters in capturing Ned and beheading him. Catelyn’s presence in King’s Landing would have made it different for Ned, she would have helped him escape to Winterfell and reunite with his family.
Catelyn would have liked Jon Snow and would never have any disdain against him. Petyr would not have owned a brothel if he had married Cat. Maybe at some point of time, his love for her would have wore off, but the thought of losing her would have hurt him very badly that he would have done everything in his power and beyond to keep her safe and with him.