Her strong language was in response to the picketing of family planning clinics across Scotland last month, which led to some temporary closures.
The move also reflects fears that Scotland could face an extension of protests to the levels seen in the United States, where people working in clinics have had their personal details published on the Internet. In the past six years, seven people have died and 12 abortion clinics have been bombed in America by extremists.
The police have already advised workers at some Scottish clinics how to guard against the possibility of attacks at home or at work.
Ms Deacon announced a big expansion in the number of clinics offering sexual health advice yesterday, in an attempt to create so many outlets for information that pro-life demonstrators would be unable to protest outside them all.
She said: "There are groups with strong views, using extreme methods to articulate what they believe... I will give them notice. I will not tolerate a situation where people, particularly women and young people, have to walk the gauntlet of often abusive demonstrators simply because they want to exercise their right and need to attend a clinic.
"I give these groups notice today - back off. Do not promote your view in proximity to these facilities. I want you to withdraw voluntarily and I want you to do so immediately."
But Ms Deacon's ultimatum was rejected by Jim Dowson, co-ordinator of Precious Life Scotland, which has organised the recent protests north of the border. Mr Dowson said: "The rise in teenage pregnancies and teenage abortions can be directly attributable to Ms Deacon's attitude in brainwashing our youth that sex is indeed a pastime without any responsibility. Ms Deacon is scared of people like us showing the truth of this loose and free and easy lifestyle. The truth is dead babies and that is why she is determined to shut us up.
"Considering we have never had a violent incident in the pro-life movement in Scotland in 32 years, I do not believe the people of Scotland will tolerate what she is saying. She will not deter us and will not put us off."
Legislation is likely to be a last resort but Ms Deacon, who was speaking at the annual conference of the Faculty of Public Health Medicine in Edinburgh, said clinic staff should "not hesitate" to seek legal redress through the 1997 Protection from Harassment Act where appropriate.
"If this does not solve the problem satisfactorily, then I will consider what more should be done," she said.
There were about 9,000 unwanted teenage pregnancies in Scotland last year and 2,000 new cases of sexually transmitted diseases among teenage girls - statistics Ms Deacon called "staggering".Reuse content