In the wake of the controversy over the safety car restart at the Abu Dhabi Grand Prix that has left Hamilton disillusioned with the sport, he and his team boss Toto Wolff elected to miss the FIA end-of-season gathering in Paris on Thursday night.
Hamilton's absence, as runner-up in the drivers' championship, appears to be a breach of F1's Sporting Regulations.
Article 6.6 of the rules states: "The drivers finishing first, second and third in the Championship must be present at the annual FIA Prize Giving ceremony."
One of Sulayem's first responsibilities since taking over will be to judge whether or not Hamilton's absence from the event goes against the regulations.
Speaking about the situation in his first official press conference at president, Sulayem said: "At the end of the day, rules are rules.
"So we are going to look into the side of where the technical rules are there to be employed and was he in breach? I have to look into it."
He added: "Of course, we'll have to be also be following our rules. But in the meanwhile, it doesn't stop us from making a champion feel good about the sport, you know.
"It's easy to be nice to people. And it is cheap to be nice. And it's also to motivate people. But definitely, if there is any breach, there is no forgiveness in this."
Sulayem said that critical for the FIA to understand was whether or not Hamilton did break the regulations, as he admitted that the world champion was 'broken' by the events of the weekend.
Asked to clarify if he could really rule out any "forgiveness", Sulayem said: "Forgiveness is always there, but rules are rules. We look at the rules.
"And I always say: rules are not made. A human made them...and they can be improved and changed by humans. So the rules are there to be improved.
"I know that Lewis is really sad about what happened and one word I would say is he's broken. But we have to look if there was any breach.
"I cannot [say for now]. It's just a few hours now I've been a president, and I've just started giving answers without going back to the facts."
If Hamilton is found guilty, then he could face a financial penalty for his absence rather than any sporting sanction.