I have never read or heard of a person who believes Calvinistic doctrine talk about "praying the sinner's prayer." The idea of "praying the sinner's prayer" seems to have originated with Charles Finney, who wasn't a Calvinist be any stretch of the imagination.
Actually the idea of a sinners pray goes back just a bit more than that. Paul suggested that here
Rom 10:9 that if you confess with your mouth Jesus as Lord, and believe in your heart that God raised Him from the dead, you will be saved;
Rom 10:10 for with the heart a person believes, resulting in righteousness, and with the mouth he confesses, resulting in salvation.
and here
Rom 10:13 for "WHOEVER WILL CALL ON THE NAME OF THE LORD WILL BE SAVED."
Rom 10:14 How then will they call on Him in whom they have not believed? How will they believe in Him whom they have not heard? And how will they hear without a preacher?
and here
Eph 1:13 In Him, you also, after listening to the message of truth, the gospel of your salvation—having also believed, you were sealed in Him with the Holy Spirit of promise,
Even in the OT we can see this
Isa 45:22 "Turn to Me and be saved, all the ends of the earth; For I am God, and there is no other.
and again here
Jer 17:7 "Blessed is the man who trusts in the LORD And whose trust is the LORD.
So you can see that it is not something new but what we do see as a new idea is what comes from Augustine and Calvin that man has nothing to do with his salvation. You were either picked out B4 creation or you are lost. This view is called divine determinism and has been codified in the DoG/TULIP.