Shaw was made to train on his own earlier this week during United's pre-season tour of the United States, with Van Gaal claiming the close-season signing from Southampton was not at the fitness levels he demands.
The 19-year-old made a surprise substitute appearance in the 0-0 draw with Inter in the International Champions Cup on Tuesday, after Reece James was taken off injured.
Shaw conceded that Van Gaal was right in his assessment and admitted he must do more to meet the requirements of playing for a club of the stature of United.
"It's something I totally agree with," Shaw said. "Playing in this formation [3-5-2] you really have to be fit. I am fit but not at his [Van Gaal's] high demands. I know I have got to get fitter to get up and down.
"That's something I am going to work on, and tonight in 45 minutes I felt really good, so I am looking forward to keep on progressing until the start of the season.
"I didn't come into United thinking it's going to be much harder and that's something I made a mistake on. I thought it was just going to be the same but United is the biggest club in the world and there are world-class players in training so it's something that is going to push me on.
"That's something I need to keep me pushing on and this is only going to help me in the long run.
"Pre-season you are always going to try and get your fitness up so I am not too sure but I played 45 minutes and I felt great. I could have played on. It's a progression from the last game I played so hopefully now the main focus is getting fitter and fitter and I'll be flying by the start of the season.
"I wasn't meant to play today because me and the staff decided it would be better to push me harder in training, but I didn't have any problems and I feel good."
Van Gaal was impressed with Shaw's performance against the Serie A outfit, adding: "I think he did very good tonight.
"He's working very hard. He was lucky today because I was going to play our friend Reece James, but he was a little bit injured.
"And so I asked Shaw to play, in spite of a very heavy training session this morning."
