No - the little bit of upset the west is feeling absorbing Syrian refugees should be a wake-up call that we can't just wall ourselves off from other peoples' problems.
Governments all over the first world are clearly more resistant to accepting immigrants than a few years ago.
One can definitely argue that for moral reasons we shouldn't wall ourselves off from other peoples problems, but the first world certainly does have the capacity to try and it seems that it also has the political desire to do so. If you ignore the words of major governments but look at their actions, we are seeing investments in capabilities to do just that, both military capabilities and economic ones - as automation replaces offshoring, the first world is now agains slowly getting less dependant on third world labor.
Please name a single first world nation-state that would be either submerged under water or have their population decimated (i.e. an expected casualty rate of 10%).
I believe that even Netherlands would stay above water by continued investment and extension of their current water control practices. For the first world, any coastal areas that actually do become underwater will likely simply result in an expensive migration, with actual deaths from storms/etc measured in the hundreds at most.
Countries like Bangladesh are in a different position, for them it makes all sense to reduce emissions, but they can't change much.
The expected worst case change in the first world is not impactful enough to warrant major changes in the country regime; if huge reductions of emissions are more expensive than some resettlement (and, generally, won't be enough to prevent the need of resettlement anyway), then we don't reduce global emissions but invest in mitigation of local consequences.
And actually, one answer to "who needs nation-states when they're submerged under water and their populations decimated?" is that the first world would actually want their nation-states to be strong and shield their people from the consequences of other nation-states becoming submerged under water and their populations decimated.