Step 1: After India became independent in 1947, Lord Mountbatten was the first Governor General of independent India (August 1947 – June 1948).
Step 2: After his tenure, C. Rajagopalachari became the first and only Indian to hold the position of Governor General of India (June 1948 – January 1950).
Step 3: The post of Governor General was abolished when India became a republic on January 26, 1950, and Dr. Rajendra Prasad became the first President of India.
Step 4: Other options:
- Dr. Rajendra Prasad was the first President, not Governor General.
- Jawaharlal Nehru was the first Prime Minister of India.
- W.C. Banerjee was the first President of the Indian National Congress (1885).
Thus, the correct answer is $\mathbfC.\ Rajagopalachari$.