Concept:
Digital India is a flagship programme of the Government of India aimed at transforming India into a digitally empowered society and knowledge economy.
Its three main objectives are:
• Digital infrastructure as a core utility for every citizen.
• Governance and services on demand.
• Digital empowerment of citizens.
Step 1: Recall the launch of Digital India.
The Digital India Programme was officially launched on:
\[
\boxed{1^{\text{st}} \text{ July, 2015}}
\]
at the Indira Gandhi Indoor Stadium, New Delhi.
Step 2: Identify the person who launched it.
The programme was launched by the then Prime Minister:
\[
\boxed{\text{Narendra Modi}}
\]
with the vision of promoting digital governance and expanding internet connectivity across the country.
Step 3: Compare the given options.
• Rajiv Gandhi -- Introduced computerization initiatives but did not launch Digital India.
• Manmohan Singh -- Not associated with the launch.
• Narendra Modi -- Officially launched Digital India in 2015.
• A.B. Vajpayee -- Promoted IT development but did not launch Digital India.
Step 4: Final conclusion.
Hence, the correct answer is:
\[
\boxed{\text{Narendra Modi}}
\]