Both Mr Carlaw, the interim Scottish Conservative leader and Scottish Labour leader Richard Leonard, raised the issue of the mesh scandal with the First Minister, after reports that the US surgeon Dr Dionysios Veronikis had withdrawn his offer to come to Scotland.
Dr Veronikis, who has developed techniques allowing the full removal of mesh implants, has written to Health Secretary Jeane Freeman - who had pledged to bring him to Scotland to help some of the 600 mesh-damaged women and to train surgeons - saying he would no longer cross the Atlantic after his treatment by senior health officials.
According to Mr Carlaw there was a belief that senior "powerful people" in NHS Scotland were attempting to "frustrate" Dr Veronikis's visit and demanded Ms Sturgeon intervene.
In an increasingly emotional question and answer session, he describing mesh as the "greatest medical scandal of modern times".
His voice breaking, he also pleaded with Ms Sturgeon to meet the affected women, and said his constituent Lorna Farrell had her life "transformed" after having her mesh removed by Dr Veronikis.
He said: “During this decade-long scandal many of the affected women feel that they have been unable to meet and discuss their experience directly with the First Minister.
“They feel that the urgency of their situation needs the direct support and engagement of the head of their government.
"Will she agree to meet the affected women directly, listen to them and give them the personal attention of the First Minister to get their lives sorted?"
Ms Sturgeon replied: "Yes I will.
"I want to make it very clear to the women affected that this does have my attention, and it has the close personal attention of the health secretary.
"I obviously understand the deep emotion that many feel about this, the women affected and those who have contact with them."
Mr Carlaw, who has been campaigning to help mesh victims for years, also alleged there was a medical "conspiracy" against bringing Dr Veronikis to Scotland.
It was reported at the weekend that Dr Veronikis did not believe officials or surgeons involved in mesh removal were ever serious about bringing him to Scotland, despite a pledge to do so by health secretary Jeane Freeman.
He said: “I am not a politician, I’m not a civil servant, I’m a doctor and do not have the time to play games. In good faith, I offered to come to Scotland for a month to operate on women and train surgeons.
“I made myself available at every stage but after months of discussions I no longer believe officials or surgeons in Scotland ever seriously tried to bring me to Scotland. I do not have time for interminable discussions and feel I must now withdraw my offer.”
Today Mr Carlaw said that it appeared the surgeon's visit had been called off due to "a co-ordinated attempt to block him by powerful people within the NHS and the medical hierarchy."
He added: "The clear suspicion of many is that there is a professional and institutional campaign to frustrate Dr Veronikis' involvement. It's a view of many that establishment figures in the NHS are trying to protect their own back."
Mr Carlaw said he had spoken to Dr Wael Agur, who was previously on the Scottish Government’s review group on mesh, who had confirmed to him that "surgeons here felt deeply threatened by Dr Veronikis’ offer to visit Scotland. No doubt there is a professional conspiracy against his visit.”
A letter from Dr Agur states: "Dr Veronikis’ visit will inevitably expose the inadequate local surgical skills.
"The surgeons blocked issuing a contract for him in NHS Greater Glasgow & Clyde Health Board. Such contract was essential for him to obtain a temporary registration from The General Medical Council to practice medicine / operate in Scotland.
"Instead, the surgeons asked government to fund a tour around 2-3 US centres, after which they will choose which surgeon to invite to Scotland. It’s arrogance, in my opinion, and will lead to waste of resources on a useless tour."
The letter also says: "The surgeons suggested another US surgeon instead: Harold Goldman - who is one of the most prominent proponent of the continuing use of mesh. In addition, he promotes partial (rather than total) mesh removal, the complete opposite of Dr Veronikis. Inviting Dr Goldman would undoubtedly support the local surgeons in their efforts to reintroduce mesh procedures in Scotland."
Mr Carlaw added: "If there is a professional conspiracy against Dr Veronikis and his proposed work in Scotland, that is an outrage. It’s something in which the First Minister must intervene."
Nicola Sturgeon stressed that the government's two priorities were to halt new mesh procedures, and a moratorium had been introduced, and to "ensure that women who have suffered complications from mesh procedures in the past receive the care and treatment they need and are entitled to, and it's something the Scottish Government takes very seriously."
She added: "I have enormous sympathy with the sentiments he expresses today.
"The Scottish Government wants Dr Veronikis to come here and that remains the case. For him to be able to treat patients here however, certain General Medical Council requirements have to be met and the Scottish Government has no discretion to waive those.
"One is a contract of employment from the NHS which necessitates clinicians from here going to see him in the United States. We had hoped that would happen in August but due to clinical commitments here that had to be postponed, but clinicians will go to the US next month and remain willing to meet with Dr Veronikis if he agrees to reconsider his position."
She added: "Jackson Carlaw talked about efforts on behalf of senior influential people, I am not aware of any such efforts, and it would not be acceptable for anyone in the medical community here to be seeking to block that.
"If there's evidence I would certainly want to see that and take action around it.
"I want, everybody wants, patients to have the treatment they need, that is considered to be clinically right for them, but also that they have confidence in and can be assured about the efficacy of. I want to make sure that women get that treatment but I do not underestimate at all the suffering, stress, pain and anxiety that many women have experienced."
And at First Minister's Questions it was also revealed that Dr Veronikis had written to Labour MSP Neil Findlay outlining his concerns about coming to Scotland - and backing claims by some of the women affected, that they had been misled about the operations they had received.
“The Scottish mesh injured women are vindicated in what they presented to Minister Freeman in March 2019,” the surgeon wrote. “What has been recorded in their medical records as a ‘full removal’ was not. It was a partial removal…”
Mr Leonard asked the First Minister: “Why have these women been misled?”
“Dr Veronikis offered these women the first glimpse of hope that they might get their lives back. The world leading, pioneering surgeon who the Health Secretary invited to come here now feels that the officials and senior surgeons in Scotland, working for our NHS, accountable to your government, obstructed this course of action.”
He also cited the case of a woman who contacted him about her six surgeries including a hysterectomy “to rid me of the daily pain” as a result of a mesh implant.
The woman wrote that she feared losing her job because of the pain, and said: “I was deeply upset to learn yesterday that Dr Veronikis has withdrawn his offer of support because no one in Scotland has fully taken control of the situation. This was possibly my last opportunity to have a pain free life.”
Ms Sturgeon said: "As First Minister I apologise to any patient who suffers on the NHS. That is something people have a right to expect.
"There is a long history in terms of mesh procedure. The actions we have taken on the moratorium to halt mesh procedures is an indication of how seriously we take this issue.
"Let me stress again that it is my desire to enable him to come here and to let patients have access to his specialism here, rather than having to travel."
On the failure of women having full mesh removals, despite being told they had, she said: "It is a point that concerns me deeply and we are determined to get to the bottom of it. I can't stress enough our determination... to bring an end to the pain and suffering these women are experiencing."