NEW DELHI—India's space agency on Monday launched five foreign satellites, reflecting its ambition to get a bigger cut of the world's $300 billion annual space business.

The launch also became a vehicle for India's new Prime Minister, Narendra Modi, to promote his regional political agenda--namely, to resuscitate India's influence as a regional power and to project himself as a champion for neighboring countries, where China has expanded its footprint. In a speech praising the launch, Mr. Modi called on Indian scientists to...