His exact intentions remain unclear but the US is reportedly considering a military response.
The BBC's North America editor says a US strike could be imminent.
US Secretary of State Rex Tillerson has said Bashar al-Assad should have no role in a future Syria.
His comments signal an apparent U-turn for the US - only last week the US ambassador to the UN, Nikki Haley, said Washington was no longer prioritising the removal of the Syrian president.
Measures reported to be under discussion between the Pentagon and the White House include the targeting of Syrian radar using cruise missiles launched from US ships.
Dozens of people, including at least 27 children, are reported to have died following an attack on the town of Khan Sheikhoun in Idlib province early on Tuesday.
Syria denies its forces were to blame and is supported by its ally, Russia.
Military action looks certain: Jon Sopel, BBC North America editor
It is a clear and pretty dramatic shift. A month ago, Bashar al-Assad in the eyes of the US was part of the solution, considered useful in the fight against so-called Islamic State.
Then yesterday, President Trump said his position had changed, that the Assad regime had crossed many lines. The implication was that there could be military action.
But given everything that's been said in the past 24 hours, I would say military action looks certain and could be imminent. We could wake up tomorrow morning and find out the Americans have taken action.
Cast your mind back to what President Trump said about Barack Obama, when the then president said a red line had been crossed and he did nothing about it afterwards.
He heaped derision on President Obama. If Mr Trump were not to act now, he would look weak and he wouldn't want that.
Russia's deputy UN envoy warned of "negative consequences" of any US military strike on Syria.
"All the responsibility... will be on shoulders of those who initiated such doubtful and tragic enterprise," Vladimir Safronkov said.
Addressing reporters on his way to Florida to meet Chinese President Xi Jinping, Mr Trump said: "I think what Assad did is terrible. I think what happened in Syria is a disgrace to humanity and he's there, and I guess he's running things, so something should happen."
Also on Thursday, Mr Tillerson said: "Assad's role in the future is uncertain and with the acts that he has taken, it would seem that there would be no role for him to govern the Syrian people.
"We are considering an appropriate response (to the) violations of all previous UN resolutions, violations of international norms."
He said "steps were under way" to organise an international coalition to remove Mr Assad.
UN Secretary-General Antonio Guterres, asked if he was concerned about possible US military action in Syria, said his priority was to ensure accountability for the deadly gas attack.
He said he could not comment on "things that have not yet happened".
Evidence has mounted that the victims of the attack in Khan Sheikhoun were killed with a nerve agent such as Sarin.
Syria has denied dropping chemical weapons from the air while Russia has argued that the mass poisoning was caused by an air strike on a rebel weapons dump where chemical weapons were being stored.
But the claims have been viewed with scepticism from the US and its allies.
Syria says it would only accept a UN investigation into the incident if a list of conditions are met.