Arsenal boss Arsene Wenger defended Walcott for making a "non-offensive" gesture towards Tottenham supporters after the England forward was pelted with missiles as he was carried off towards the end of the third-round tie at the Emirates Stadium.
The Gunners booked their place in the fourth round draw after Santi Cazorla drilled in a fine opening goal on 31 minutes, before Tomas Rosicky doubled the lead on the hour with a well-taken finish after robbing Danny Rose on the halfway line and sprinting clear.
There was, however, a controversial end to what had been a fiercely contested north London derby when Walcott was taken off on a stretcher with a knee injury.
Walcott was treated by medical staff onto a stretcher, before being taken around behind the Arsenal goal, in front of the Spurs fans - some of whom threw objects, which appeared to include coins and plastic bottles.
Walcott signalled a "2-0" gesture with his hands, before donning an Arsenal scarf as he was carried on around the stadium.
The Football Association will wait until receiving the match report of referee Mark Clattenburg before deciding whether the incident warrants investigation.
Sherwood revealed in his post-match press conference he had not witnessed the incidents.
He said: "Someone just mentioned it to me in there. I didn't see it.
"Is that what he did? Well, listen I don't know Theo, I don't know him as a lad. It could have been a bit of banter, possibly.
"If they (Tottenham fans) did then it's a problem, isn't it? If they didn't then it's not a problem."