NEW DELHI — A deadly attack on tourists in Indian-controlled Kashmir has again moved India and Pakistan closer to war as the two rivals downgraded diplomatic and trade ties, closed the main border crossing and revoked visas for each other’s nationals.

The Pakistani government said that if India followed through on a threat to block rivers it would take it as “an act of war.”

Indian officials said the army had a brief exchange of fire with Pakistani soldiers along their highly militarized frontier in the Himalayan region of Kashmir, as the nuclear-armed rivals ramped up tit-for-tat diplomatic offensive following a deadly attack on tourists in the disputed area.