Well, it would be an act of war if the US wanted to see it that way. However consider the following.
1) South Korea may be reluctant to agree to the USA kicking off a major war on the Korean peninsula particularly as Seoul is within artillery range of the border. The DPRK has a ridiculous amount of tube and rocket artillery pointed at the South Korean capital. I suspect the RoK would not like having to rebuild most of its modern infrastructure or cope with the tens of thousands of casualties that would result.
2) North Korea has a nuclear weapons program (thank you Dr Khan), how many of them they have and how reliable and deliverable they are is a worrying question. The Japanese may also be reluctant to agree to the US using their bases in Japan for ops against the DPRK as there is a risk Tokyo could be flash fried.
3) Apart from the Norths nuclear program they also have a significant chemical capability, they also have a lot of long range artillery and ballistic missiles which could be (are?) fitted with chemical agent warheads.
4) China may decide to resolve the issue by either engineering a coup in the DPRK (there have been suggestions they have tried this previously, but to be honest the internal politics are so opaque who'd know) which would put the US in quandry, do they attack a country which now has a new regime and backed by China?; Alternatively, the Chinese may just go for broke and eliminate the DPRK themselves, less likely maybe but possible. In the long term this may result in the RoK becoming either non-aligned or actively cooperating with China. Having the outdated DPRK military across the border is one thing, having the PLA sitting there becoming increasingly state of the art is another.
5) The cost of a war against the DPRK in both men and materials would be horrendous. The likely casualty rates just on the US side would be horrific as the DPRK is literally dug in with numerous hardened artillery sites, bunker complexes, underground airbases (like something off thunderbirds) and millions of men under arms or in militia units. The US would literally be fighting for every city, town, village and hamlet. If it went nuclear, the losses would be even worse.
Given the above it is possible the US would settle for some sort of (even more) punitive sanctions or very limited military action such as a Tomahawk strike against a number of NK military installations.