Kitson fell out with boss Pulis after making a rude gesture towards the subs' bench when he was taken off during the Potters' 6-0 thrashing at Chelsea in April.
He was then made to train with the youth team - before a transfer deadline-day move to Pompey.
Kitson said: "My plan was to stay at Stoke and get involved there and I trained hard every day to make sure I was ready to play.
"As the window went on it became more apparent that was not going to happen.
"I am 30 years old and Portsmouth offered me the opportunity to play.
"My departure from Stoke was amicable and I wish them all the best although I don't think Tony Pulis and I will be exchanging Christmas cards."