The Americans, who went on to win the trophy by 14 1/2 points to 13 1/2 after a spectacular display in Sunday's singles, were enraged when European pair Pettersen and Charley Hull, 19, beat Alison Lee and Brittany Lincicome in the morning fourballs.
With their match all square, rookie Lee mistakenly believed her 16-inch putt at the 17th had been conceded by her opponents and picked the ball up, leaving the referee no choice but to award the hole to the Europeans in the women's version of the Ryder Cup.
Most of the American ire was aimed at the experienced Pettersen who was initially unrepentant when she was quizzed by reporters at St Leon-Rot about the incident on Sunday evening but has now had a change of heart.
"I've never felt more gutted and truly sad about what went down," the 34-year-old Norwegian said on Instagram on Monday.
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"I am so sorry for not thinking about the bigger picture in the heat of the battle. I was trying my hardest for my team and put the match and the point that could be earned ahead of sportsmanship and the game of golf itself."
Pettersen, the highest-ranked player in the European team, said she had personally sought out Inkster on Monday.
"To the U.S. team you guys have a great leader in Juli who I've always looked up to and respect so much," said the world number eight.
"Knowing I need to make things 'right', I had a face to face chat with her before leaving Germany this morning to tell her in person how I really feel. I wanted her also to know I am sorry.
"I hope in time the U.S. team will forgive me and know I have learned a valuable lesson about what is truly important in this great game," added Pettersen, a regular on the American circuit.
"To the fans of golf who watched on TV, I am sorry for the way I carried myself. I can be so much better and being an ambassador for this great game means a lot to me.
"I wish I could change Sunday for many reasons. Unfortunately I can't," said Pettersen, a veteran of eight Solheim Cup campaigns.
(Editing by Martyn Herman)
