Step 1: Context.
On 15th August 1947, India became independent from British rule. With independence, India required a Governor General to act as the head of state until the constitution was fully implemented.
Step 2: Appointment.
Lord Louis Mountbatten, who was the last British Viceroy of India, was appointed as the first Governor General of independent India. He continued in this role until June 1948.
Step 3: Successor.
After Mountbatten, C. Rajagopalachari became the first and only Indian Governor General of India.
Step 4: Conclusion.
Thus, the first Governor General of independent India was Lord Mountbatten.
\[
\boxed{\text{Correct Answer: Lord Mountbatten}}
\]