His name was daubed on a wall in the Newell Road area of Dungannon along with a target mark.
It came just hours after the Stormont Executive agreed plans to roll out a mandatory vaccine passport.
It also appeared hours after a 44-year-old man was remanded in custody on charges which include making threats to kill Mr Swann as well as harassment and improper use of electronic communications.
The Police Service of Northern Ireland (PSNI) confirmed it received a report of criminal damage at the Corrainey Park area of Dungannon.
Superintendent Michael Simpson said: “Sometime between 11pm on Wednesday November 17 and 7.30am on Thursday November 18, graffiti was painted on a wall of a property in the area.
“Enquiries are continuing and police would appeal to anyone who witnessed the incident to contact them on 101, quoting reference number 286 18/11/21.”
Ulster Unionist MLA Rosemary Barton condemned the graffiti.
“The vast majority of people in Northern Ireland recognise the sterling job that Robin Swann has done as Health Minister throughout the pandemic,” she said.
“Unfortunately, it appears that a few individuals are unable to see what the rest of society can.
“Two days ago he had to make a complaint to the police about threats made against him and his family and today we have learned that someone has taken it upon themselves to spray a threat on a wall in the Newell Road area of Dungannon. I genuinely pity the type of mind that would do such a thing.”
DUP MLA Deborah Erskine also condemned the graffiti.
“I am absolutely appalled to see this graffiti. This is a sinister threat towards a minister with a young family, who is simply trying to do his best,” she said.
“There are those who may disagree with the political decisions made yesterday but this is an appalling incident and should be wholeheartedly condemned.
“Public discourse and vile attacks must be outrightly condemned by everyone.”
Mr Swann, as well as Northern Ireland’s Justice Minister Naomi Long, was also targeted with abuse on social media on Wednesday evening.
Northern Ireland Secretary Brandon Lewis was among political leaders who expressed solidarity with Ms Long.
“The vile misogynistic abuse that has been directed towards her since yesterday’s vote is disgusting and has no place in this echo chamber or outside of it,” he tweeted.
Sinn Fein vice-president Michelle O’Neill tweeted: “The misogynistic abuse and harassment of women online is deplorable.
“Trolls attacking women on their appearance are cowards. Women are mothers, sisters and daughters.
“Our families should not be exposed to this vile abuse. The abuse must stop now.”
Ulster Unionist leader Doug Beattie said the abuse directed at Ms Long and Mr Swann was “utterly disgraceful”.
“You can have a point of view, that’s fair, but you’ve no right to abuse anyone on here or anywhere else. #StopTheAbuse,” he tweeted.
Sinn Fein MP Chris Hazzard tweeted: “You have to be a cowardly weasel to launch the type of attack on a female public representative that Naomi Long has suffered today on Twitter. Utterly pathetic Solidarity with @naomi_long.”
SDLP leader Colum Eastwood urged social media companies “to stop allowing thugs to turn their platforms into cesspits”.
Ms Long’s husband, Michael, a councillor in Belfast, thanked all those who had expressed support.
“Social media is a sewer and full of non-entities seeking attention. Thanks to many people who have expressed support from across the spectrum for Naomi tonight,” he tweeted.
The Executive was divided over mandatory vaccine passports, with the DUP voting against them.
The other four Executive parties backed a proposal from Mr Swann that will see legal enforcement of Covid-19 certification entry requirements for nightclubs, bars, restaurants and a range of other settings from December 13.