Irish international Walters had a truly miserable time of it on Saturday, heading in two own goals and missing a penalty as the Potters were beaten 4-0.
"I don't think anybody will worry about Jon Walters," Begovic said.
"His character and personality won't be affected. It was a couple of unfortunate incidents for him. I take the blame for the second goal because I probably should have come for the ball, but seeing him there I decided not to."
Pulis is set to make a number of adjustments to his first XI tonight.
Goalkeeper Thomas Sorensen looks set to come in for Begovic once again, while striker Michael Owen will be hoping to be back involved after starting the initial meeting with Palace but then missing out on a place in the squad for the Chelsea game.
A Dundee United spokesperson said: "We won't stand in the way of anyone furthering their career."
Bournemouth defender Tommy Elphick is determined to create history as his side host Wigan tonight. Eddie Howe's side matched a club record which had stood for 30 years on Saturday as they extended their unbeaten run to 18 games with a draw against Swindon.
"It's a marvellous feat and it's going to take another really good team to do it again," he said. "We won't be looking to stop at 18. We will be looking to extend it for a lot longer."
"We started sluggishly against Bolton and I don't think we can afford to do that, but against West Ham we started strongly and the most pleasing aspect for me was that we kept it going," he said.
"For the most part, we were terrific."
FA Cup, third-round replays (7.45 unless stated): Birmingham v Leeds Utd; Bournemouth v Wigan; Brentford v Southend; Leyton Orient v Hull; MK Dons v Sheff Wed; Stoke v Crystal Palace; Sunderland v Bolton; Blackpool v Fulham, 8.0; West Brom v QPR, 8.0.