What is the ruling on one who doubts in his four-rak'ah prayer whether he completed the recitation of Al-Fatiha or not, and whether it is the third or fourth rak'ah, so he based his action on the lesser in both cases, and then performed the two prostrations of forgetfulness (sujud al-sahw)? Is his action correct?

What you did is incorrect. The correct approach is for you to rise from your bowing (ruku') to recite Al-Fatiha, then perform one rak'ah, and prostrate for forgetfulness (sujud (definition) sahv) before the salutation (taslim (definition)). This is if you are not prone to excessive doubt (waswasa). If you are, then ignore these whispers, and no prostration for forgetfulness is required of you.

Your mentioned prayers do not need to be repeated due to your ignorance of the ruling.