A pass over the top by Fabian Schar was poorly dealt by Spurs goalkeeper Hugo Lloris, who bumped into Wilson before the Magpies No.9 collected the loose ball and fired into an empty net.
The home side complained to referee Jarred Gillett but after a VAR check, the goal stood and United went 1-0 up front on the half-hour mark.
“There’s nothing wrong,” Gallagher said. “Lloris goes into him, I’m not sure where Callum Wilson can go. I think the referee is absolutely right to give a goal and VAR concurred.”
Former Spurs and England goalkeeper Paul Robinson did not agree, however.
Robinson said: “Listen, I think it’s soft by the goalkeeper but you look at Wilson’s arm, Hugo dives, the contact wasn’t strong enough to send him to ground but I see Wilson’s arm come around Lloris.
“If that’s the other way around and Wilson chests that around Lloris and Hugo puts his arm around him, that’s a penalty in the other direction.”
Gallagher responded: “You said yourself, I don’t think he’s done too much wrong. I don’t think there is enough evidence to give a foul against him. Once the referee gives the goal on-field, the VAR isn’t going to get involved. It’s got to be a very big foul to give it.”
United’s one-goal advantage quickly extended to two via Miguel Almiron, and Eddie Howe’s side held out for a big three points despite Harry Kane pulling a goal back in the 54th-minute.