Depay had a big part to play in both Stoke goals, his weak header back to David de Gea allowing Glen Johnson in to set up Bojan and his failure to close down Marko Arnautovic didn't help for the second.
The Stoke defeat meant United have gone seven games in all competitions without victory and is the first time they have lost four games in a row in the same season since 1961.
United have slipped down to sixth in the Premier League table as a result of their desperately poor form.
But Shearer - who once famously turned down a move to United - says van Gaal is not being helped by terrible performances from individuals like Memphis Depay.
'He could have done better'
And the BBC pundit, who scored 313 goals in 622 career games for club and country, told Match of the Day Depay could learn a thing or two from Stoke star Marko Arnautovic.
"Anyone can make mistakes, and it was a mistake, it was a poor header from Depay he just didn't get the connection at all," Shearer said on MOTD.
"Tracking back, I don't think there's enough effort from Depay, he gives up far, far too easily.
"I think he can do better here (with the Arnautovic goal), I think he can close the shot down. I think if he wants to get hit with the ball and is prepared to then he will do.
"In contrast watch Arnautovic get back and help his teammates, he wants to fight and he wants to scrap for his teammates.
"And then when you do that, you get the rewards like Arnautovic did there."
Arnautovic by contrast cost Stoke just £2m.
Both operate on the left and both have had seasons that are poles apart.
Austrian livewire Arnautovic has been one of the stars of the Premier League season as Stoke look to assert themselves as a top eight side.
Depay has had a miserable start to life in Manchester, scoring just twice in 13 games for the Red Devils.