In a speech to parliament later, he said he would address Delhi's bid for a permanent UN Security Council seat.
The US president is on a 10-day Asian tour designed to boost US exports.
He is also due to visit Indonesia, South Korea and Japan.
We have not only an opportunity, but a responsibility to lead”
End Quote Barack Obama US President
President Obama told Monday's news conference the two leaders had been discussing the situation in East Asia, which many believe is essentially a focus on China.
Mr Singh said Washington and Delhi had decided to "accelerate the deepening of ties to work as equal partners in a strategic relationship".He said he and Mr Obama had agreed protectionism was detrimental for both countries, and that India was not in the business of stealing American jobs.
Both sides would expand co-operation on space, civil, nuclear and defence matters, he added.
Mr Obama announced an initiative between both sides to improve security at airports, ports and borders.
Taking questions from journalists, Mr Obama stopped short of committing the US to intervene in India's long-standing dispute with Pakistan over Kashmir.
Washington "cannot impose a solution to these problems" in the Muslim-majority Himalayan border region, he said.
'Terror-induced coercion' Mr Singh said India was committed to resolving all problems with Pakistan, but Islamabad should move away from "terror-induced coercion".
The US leader's three-day visit to India is being viewed by many in Pakistan as a snub to the Islamic republic.
Summing up their ties, Mr Obama said: "As the world's two largest democracies, as large and growing free market economies, as diverse, multi-ethnic societies with strong traditions of pluralism and tolerance, we have not only an opportunity, but also a responsibility to lead.
"And that's why I believe that the relationship between the United States and India will, in fact, be one of the defining partnerships of the 21st Century."
Mr Singh said the two countries' relationship would be "defining and indispensable" for the coming decades.
On his third and final day in India, Mr Obama received a ceremonial welcome at the Indian president's palace, and laid a wreath at a memorial to Indian independence leader Mahatma Gandhi.
At the weekend, he visited the scene of the 2008 attacks in Mumbai and criticised the pace of Pakistan's fight against militants within its borders.
He also announced $10bn (£6.2bn) in new trade deals with India.
Trade between India and the US was worth about $40bn in 2008 - still significantly less than US trade with other partners like China and Europe.