The former finance secretary stunned his parliamentary colleagues when he resigned his position just hours before he was due to deliver the Scottish Government's budget for 2020.
Mr Mackay, 42, was revealed to have sent hundreds of unsolicited messages to a 16-year-old boy via social media.
The former SNP high-flyer, who has not been seen in public since the scandal broke on February 5, was suspended by the SNP but remains an MSP. Under parliamentary rules, parties cannot force individuals to resign their seats.
He has represented the Renfrewshire North and West seat since 2011.
Scottish Conservative chairwoman Rachael Hamilton said today: “It’s now almost a month since these shocking revelations came to light.
“That’s a long time for an elected MSP to be in hiding, and a decision has to be made. He cannot go on conning his constituents and shunning their needs – he has to quit now.
“We also need to know from the Scottish Government and the SNP if they have been in touch with Mr Mackay, and if he has indicated to them whether or not he intends to take his sizeable ministerial pay-off.
"Both Derek Mackay and the SNP clearly hope this matter will simply be forgotten about if they ignore it for long enough. But that won’t wash with the Scottish people.”
Last week, a Scottish local authority previously led by Mr Mackay called on him to resign as one of the area's MSPs.
At a meeting of Renfrewshire Council, elected representatives from all parties backed a resolution calling for the ex-SNP minister to stand down from his Holyrood seat.
The motion was lodged by Labour group leader Eddie Devine and seconded by Councillor Jim Sharkey but passed with no opposition from SNP councillors.
In allegations published earlier this month, Mr Mackay is said to have contacted the boy "out of the blue" and sent about 270 messages on Instagram and through Facebook.
In one of the exchanges it is claimed that Mr Mackay told the teenager he was "cute". In another the boy confirmed he was 16 and tells Mr Mackay "not to try anything".
The teenager subsequently made a statement to Police Scotland but has not made a formal complaint regarding the behaviour of the politician.
An SNP spokesman said: “#BeKind is trending for a reason. The Tories playing fast and loose with someone’s mental health status in a desperate bid to try and score cheap political points is really low.”