The US was justified in using the atomic bombs against Japan.  Using the bombs saved tremendous numbers of American lives.  In addition, while the US may not have actually considered this at the time, the atomic bombs surely saved many more Japanese lives than they took. 

When a country is at war, its first responsibility is to win the war while taking the fewest possible casualties.  Of course, there are some rules about how the enemy must be treated, but the atomic bombs did not break these rules.  While we treat the atomic bomb as if it were somehow much different and much worse than conventional weapons, it is not clear that it really was.  For example, the Americans firebombed Tokyo on one night in 1945 and killed more Japanese than were killed immediately by either atomic bomb (but not more than both combined).  It is hard to see how the atomic bomb was much worse than this sort of bombing. 

In addition, the atomic bombs were surely better than what would have happened if the US had invaded the home islands.  It is estimated that as many as 100,000 or more civilians died in the American invasion.  If that many Japanese could be killed in the invasion of a small island, how many more would have been killed in an invasion of all of the main islands of Japan?

When we look at it like this, the decision to drop the bombs is justifiable.  What the bombs did was horrible.  What would have happened to American military personnel, Japanese military personnel, and Japanese civilians in an invasion of Japan would have been worse.

How did the US justify its decision to use atomic bombs against Hiroshima and Nagasaki?

It is, perhaps, historically inaccurate to say that the US justified its decision at the time that it was made.  At the time, there was almost no controversy over the use of the bomb.  Winston Churchill reported, for example, that it was simply assumed that the US would use any weapon that it had to win the war.  The idea that the US needed to justify the decision came mainly after the war.

The justification that has typically been given is that the atomic bombs saved lives.  If the United States had had to invade the home islands of Japan, the toll in human life would have been horrific.  There are estimates that a million Americans would have been killed or wounded. The number of Japanese casualties would have been much higher.  Using the bombs to end the war quickly, then, was undoubtedly horrible for the people of the two cities involved.  However, it likely saved the lives of many more people than it killed.  This is the major justification for the use of the bomb.