Looking at the various responses, it seems many people are rationalizing the actions by
1. Americans should not assume they can go anywhere in the world without a visa.
2. The immigration officials were doing their job.
3. How can we let someone inside India without a visa?
These are valid points, but firstly we should not care whether it is an American or anyone else everyone should need a visa. If they cannot provide a visa give them reasonable time if they can claim to be able to obtain it. Deport adults immediately if need be but be more considerate with children. We cannot justify our actions because others behave irrationally. I would strongly recommend people who are vehemently arguing for the official's actions to travel with children for 18+ hrs and then comment. There is no reason to defend the parents as they committed a mistake they'll never forget but I think it would have been a better experience for all (officials included) if they could give more time only because children were involved.
