The Celtic midfielder's first international goal - Scotland s first and only of the tournament - was not enough to salvage the nation's hopes of a place in the knock-out phase of a major finals for the first time in history after a 3-1 defeat to Croatia at Hampden.
McGregor struck three minutes before half-time to cancel out Nikola Vlašić's opener but a stunning strike from Luka Modric after the break put a dagger through Scotland hearts before Ivan Perisic ensured Croatia's place in the last 16 with their third 12 minutes from time.
For Steve Clarke's side attention now turns to a tough World Cup qualification triple header in the first week of September with a home match against Moldova sandwiched between trips to Denmark and Austria.
"The lads are shattered. They put so much energy into it," McGregor said.
"The fans were amazing again so we just want to thank them for their input. We just lost to too much quality in the end. The second goal, Modric puts it in with the outside of his foot into the top corner. That's what you're up against at this level so it's a tough lesson.
"It's always nice to score and to get my first goal for my country is a proud moment, but ultimately we're disappointed that we didn't get the result we wanted.
"It's been an amazing experience. Every single one of us has learned so much. It was such a tough learning experience against some really top players, but it just gives you the hunger to get back here. We've got the World Cup in a couple of years and we want to make sure we're on that stage again. It just makes us so hungry."