The clinics affected by the bill include the Queen Elizabeth University Hospital in Glasgow, the Royal Infirmary in Edinburgh and Dundee’s Ninewells Hospital.
Within the buffer zones, it is a criminal offence to behave in ways that could influence the decisions of women and staff to access services.
Stopping women and staff from entering the clinics or otherwise causing alarm, harassment or distress will also be an offence.
Anyone who breaks the new Safe Access Zone laws could be fined up to £10,000 or an unlimited amount in more serious cases.
Police Scotland will be responsible for enforcing the legislation.
Speaking as the laws came into force, Mackay said: “This is a crucial day for reproductive rights and healthcare in Scotland.
“I hope that it will be the end of the intimidation and harassment we have seen of people who are accessing healthcare.”
She added: “Right from the first moment I saw footage of the protests, I could see how much damage they were doing and how many people were being impacted by them. I knew that I had to do everything I could to stop them.
“Over the days and weeks ahead, I will be working with the Scottish government to ensure that patients and staff know where protesters can and can’t be so that they can report any activity that is against the law.”
