Specifically, does the U.S. Military use a nuclear weapon against the monster? Some reviewers have assumed they do. Although we never see the events after the rubble envelops the camera, the answer would appear to be no, for several reasons:
1. The two explosions that take place outside the tunnel, filling it with debris, appear to be from conventional weapons. A nuclear weapon would have produced a blinding white flash followed by a much larger "vaporizing-effect" explosion. Assuming the military was attempting a direct hit on the monster, and the monster was still nearby (meaning, still in the park with Rob and Beth), then Rob, Beth, the bridge, and the camera would have been completely vaporized, leaving no record.
2. The production team got all their geography and physics right. We can assume that had they intended to suggest a nuclear weapon, they would have portrayed it in the correct way, with the above-mentioned flash. The two distinctly separate explosions reinforce the idea that the filmmakers took great care with their effects.
3. Although the soldier refers to "level[ing] Manhattan", being "prepared to let this whole area go", and "Operation Hammer Down," no one uses the adjective "nuclear". The military may have been prepared to go that far, but there is no evidence that they use nuclear weapons during the events depicted the film. And even if the military had been prepared to use such weapons they do not control the use of nuclear weapons on the ground. It would have required direct Presidential action to authorize the use of a nuclear weapon.
4. The United States government would be extremely reluctant to use nuclear weapons to eliminate a threat on American soil. The blast would kill refugees in the surrounding boroughs and the additional problem of wide spread radioactive contamination, especially from a surface burst weapon, would have been unacceptable unless it was the absolute last resort.
5. There are a variety of large air dropped ordnance that would have been used before the deployment of a nuclear weapon. These include thermobaric weapons and fuel air explosives. Both of these use extremely large amounts of chemicals that rapidly react to produce tremendous quantities of heat and a large blast overpressure. Unlike nuclear weapons they do not produce radioactive fallout and their effects are generally far more limited in scope than most nuclear weapons. The Thermobaric bombs, and extremely large high explosive weapons such as the MOAB (massive ordnance air burst [aka "mother of all bombs"]) are guided by GPS so hitting a moving target, even one as large as the Cloverfield monster would be difficult. Other ordnance uses laser guidance so they can hit any target being designated (painted) by the correct type of laser. This includes "bunker buster" ordnance, which are several thousand-pound bombs with a thick steel casing designed to allow them to penetrate several yards of heavy concrete before detonating. In addition, seeing as the B2 Bomber's free fall bombs appeared to have some effect on the creature, there is no reason to suggest that a continuous saturation bombing using conventional missiles, artillery and bombs would not also have significant effect.
6. In addition, the two explosions suggest multiple bombs falling upon Central Park. If a nuclear device was being used, it would seem pointless in dropping several bombs prior to the nuclear detonation that some people believe to be the ending. Also, multiple bombings (conventional then nuclear) would require several different jets in the air. In order to drop a nuclear bomb, a specific type of jet is required, to shield it from the EMP (electromagnetic pulse) generated from the bomb. However, the Army may have used nuclear cruise missiles or heavy artillery instead of jets.
7. For the military to "retrieve" the camera (as stated in the opening title cards) from a nuclear blast zone would mean that after the bombing, they engaged in a massive search & recovery effort, which would have meant digging through many square miles of radioactive waste and rubble. Not only is this unlikely in itself, but for them to find and single out the miraculously undamaged footage from what would have been millions of personal effects scattered throughout Manhattan is borderline impossible.
8. Were nuclear weapons used after the events depicted? That's unlikely. Again, a nuclear explosion would have destroyed the recording inside the camera. Even if the blast and fireball did not destroy the camera the electromagnetic pulse would have destroyed the recording.
9. Finally, the smoking gun of the whole debate: in an interview with J.J. Abrams in Rolling Stone Magazine, when RS asked Abrams if the pictures on 1-18-08.com suggested that the monster is killed by the Military, Abrams responded, "Yes, he's dead. Ultimately the bombs kill him." The photographs referred to on 1-18-08.com are a picture of firstly, a picture of Naval Ships and Aircraft advancing out to sea with a sinking ship in the foreground, then another photo with many ships and aircraft bombing an area of the ocean, then another picture of bloody waters, and finally a photo of pieces of the monsters carcass washed up on against the shore. None of these photographs feature any nuclear weapons, only conventional missiles and bombs.