Murray became the first British man in 76 years to win a grand slam singles title at the US Open when he beat Novak Djokovic over five sets.
The Scot's odds of becoming BBC Sports Personality of the Year have been slashed but the 25-year-old is not expecting to be made Sir Andy Murray any time soon.
He said: "A lot of my friends have been messaging me about it and I don't really know what to say. I think it should take more than one or two good tournaments to deserve something like that. It would probably be a bit rash."
The final finished after 2am British time but many people in Murray's home town of Dunblane and across the country stayed up to watch and celebrate the Scot's success.
He received messages of congratulations from Prime Minister David Cameron among others, and he admits the reaction he will get when he arrives home is something he cannot yet imagine.
Murray said: "All of that will probably hit me when I go back. It's something that will probably take a bit of getting used to. It's not something I've always been that comfortable with.
"I spoke to (coach) Ivan (Lendl) a couple of times during the year and he asked me, 'What worries you?'.
"And I said that I worry what might happen if I win a major, how my life might change, because I want it to be the same.
"He said he felt the same thing but all that happens is you get more people congratulating you and you get nicer tables in restaurants and to play on all the good golf courses for free."
Murray's US Open triumph came five weeks after he avenged his Wimbledon final defeat by Roger Federer to win Olympic gold, at the time the biggest victory of his career.
It was fitting the 25-year-old should top that just hours after Britain's Olympic and Paralympic athletes were saluted with a parade around the streets of London, attended by hundreds of thousands of people.
The only disappointment for Murray was he was unable to be part of it, but he is hoping to be able to take part in a similar parade for Scotland's athletes in Glasgow on Friday.
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The Scot hailed the effect Britain's magical sporting summer has had on the country, saying: "Being around the Olympics and seeing how the nation came together, from the public to the athletes to the press, everyone was just right behind it.
"I saw some of the pictures of the parade and it's just amazing to see how pumped everyone was. It's been amazing to be part of it.
"Sport has been this huge part of my life since I was a kid and it's been the best summer of sport in my lifetime and I'm sure in most people's. It's been so much fun and I'm just happy I was able to contribute towards it."
Murray has had countless personal messages but the one that really meant a lot to him came from a man who has foiled him at grand slams on so many occasions, Rafael Nadal.
The Spaniard missed the tournament with a knee injury that is likely to keep him out for the rest of the season, and Murray overtook his friend and rival to become world number three.
Murray said: "He messaged me after the match just saying, 'Enjoy it, I'm very happy for you'.
"I got a lot of congratulations from a lot of people but when you get it from someone that you're competing against, and he's one of the best players ever as well, it means a little bit more."
Murray celebrated with a team dinner last night, where he stuck to being teetotal, unlike the rest of the party.
This morning he was much in demand, appearing on the CBS breakfast show and heading to a photoshoot with the trophy in Central Park before attending a reception at British Consul Danny Lopez's official residence in Manhattan.
He was welcomed into the penthouse by Scottish piper Don Neil MacRitchie playing Scotland The Brave and presented with a hamper of British food and drink, including Hula Hoops, Maltesers and Hobnobs, although in true Scottish fashion Murray went straight for the Irn-Bru.
Attempting to describe his emotions, he said of his breakthrough title: "I don't know if you can ever imagine something like that. I've dreamt things and thought about it but you never get the same feeling as when it actually happens.
"I was in a bit of shock and after that you're very relieved. I wasn't able to sleep last night. I wasn't bouncing off the walls or anything, I just couldn't go to sleep, I was just sitting awake for a few hours.
"During the tournament, if I'd had an hour and a half's sleep and had to get up I would have been in the worst mood ever but I woke up and jumped out of bed at 6.30am, which isn't like me. I'm very excited but it'll probably take a few days for it to sink in."