What is the ruling of Islamic law regarding the wife remaining married to the man after discovering her deception in claiming illness and undergoing open-heart surgery, and what is the correct course of action in this situation, knowing that there is a child?

Lying is forbidden and a major sin, and a believer is not characterized by it. The Prophet, peace and blessings be upon him, warned against it and explained its dire consequences. No matter how great the sin, the door to repentance is open. If your wife repents and becomes upright, then keep her and treat her well, and strive to forget the past, in order to preserve the family. Divorce is easy, but it is difficult to bear its bad consequences, the scattering of the family, and the loss of the children. Therefore, some scholars have deemed it forbidden unless there is a necessity for it. Your statement to your wife: "She will not remain under your marital bond for another day" is not an explicit divorce, so it does not constitute a divorce unless you intended to divorce her immediately.