The First Minister said she had asked her officials to look into the matter.
She told MSPs the ban on large outdoor events would be lifted from Monday as Scotland begins to “turn a corner” in its pandemic response.
But details of her announcement were reported in advance.
Speaking in Holyrood on Tuesday, Ms Johnstone said: "Before I call the First Minister, I am aware that details of today’s Covid-19 update have been reported in the media prior to being announced to the Parliament.
"I should not be reading in the media that they have seen a draft copy of the statement before it has been made to this Parliament. This is extremely disappointing and is disrespectful to this Parliament and its elected members.
"I have spoken to the minister for parliamentary business [George Adam] about this and he has informed me that the Government will be undertaking an inquiry into the leak.
"Now I will allow this statement to be delivered today because I cannot be assured that all elected members have seen the full draft and also that it's important that the public hear the full detail.
"But before I call the First Minister, I ask for an assurance from the First Minister of her commitment to the place of this Parliament being respected by her Government and her commitment to a repeat of this situation being avoided."
Ms Sturgeon said: "I have asked my officials to look into this matter, and I can advise the chamber that the Scottish Government’s chief information security officer will be undertaking appropriate enquiries to establish if there has indeed been a leak of some of the contents of this statement in advance of me delivering it, and if so what the circumstances and the source of that are.”
She added: “Let me assure you, Presiding Officer, that I take very seriously my obligations and responsibilities and those of my Government to this Parliament, and I hope you will accept that assurance.”