The American is just one of a succession of fighters to be linked with a bout against Calzaghe, whose win at Madison Square Garden in the early hours of Sunday morning saw him stretch his unbeaten run to 46 fights.
Bernard Hopkins, who Calzaghe defeated earlier this year, and a rematch with Jones Jr have been mentioned as possible opponents, but Shaw believes he should be looking at fighting the next generation of boxers.
He added: "What good would it do for Tiger Woods to take on Arnold Palmer or Jack Nicklaus?
"If you're really the best then you always want to take the best on. Win, lose or draw, Joe is a Hall of Famer inside the ring and outside the ring.
"So why not take on Dawson who's the best?"
Several ex-boxers have urged Calzaghe to call it a day while he is still at the top with Ricky Hatton joining the calls for him to retire.
I believe there is only one more thing Calzaghe has to do - get Antonio Tarver
"There's nothing more Joe can do," Hatton told Radio 5 Live. "There's always a new contender coming through.
"But he's beaten Bernard Hopkins, he's beaten Roy Jones, he's done Madison Square Garden and Las Vegas. You can't get a better way to bow out."
Former world heavyweight champion Lennox Lewis has also urged him to call it a day.
He told 5 Live's Sportsweek: "He's beaten everybody there is to beat and doesn't have anything else to prove, so I think he should retire."
British heavyweight David Haye agreed with Lewis, adding: "Very rarely do you get a fighter with his record retiring at the top.
"I think Lennox Lewis showed you can walk away from the game and stay away from it and get out on top.
Calzaghe (right) beat Hatton to the 2007 Sports Personality of the Year
"It depends on how much money he's got. If he's got enough money to comfortably have the lifestyle he wants then he would retire and if he hasn't then he'll only be fighting for the money, which a lot of boxers do later on in their career.
"Fingers crossed he's tucked a bit away and he can have a nice lifestyle with his family."
However, American middleweight legend Marvin Hagler told 5 Live Calzaghe had one more fight left in him.
"He has beaten legends in his time and now I believe there is only one more thing he has to do and that is to get (former light-heavyweight world champion) Antonio Tarver," said Hagler.
"Then he can shut up all the critics and he will go down as one of the best fighters of all time.
"He will make the right decision. He will know, his body will tell him, if he can continue or not."
Calzaghe's former manager Frank Warren believed the Newbridge fighter would postpone plans to hang up his gloves.
"Before the fight he was saying it was his last fight and now he's saying he's not sure," said Warren, whose 12-year association with Calzaghe ended after the southpaw's victory over Hopkins in April.
"I believe he'll fight on. I would like to see him retire. He's got nothing else to prove and he's without doubt the greatest super-middleweight of all time.
"For me, he's the best British boxer since the war. His record speaks for itself.
"They say that fighters are the last people to know when it's time to give up, but I think they're the last ones to admit it, and that's what he needs to do."
Hopkins, who Calzaghe defeated in Las Vegas, and Mikkel Kessler, whom he beat in 2007, are among a small list of potential opponents for Calzaghe if he chooses to continue.
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