They had no choice but to support it. The emperor is
incredibly popular in Japan. In fact, the abdication threw a spanner in the works for Abe and the LDP wrt the constitutional changes hey wanted for allowing Japan to get involved in external conflicts if its allies are involved.
For some history, this is the second time Abe has been prime minister. The first time he had to resign. This was directly after proposing the constitutional change. Now, Wikipedia (and the western media it seems) reported that Abe resigned due to the poor performance of his economic policies. However, at the time my impression was that he resigned over the backlash to the constitutional proposals. At the time my Japanese wasn't very good, so I could definitely have gotten it wrong, but that was certainly what I understood.
Fast forward to a few years ago. Abe got back into power, instituted his Abenomics and then re-proposed the constitutional changes. It seemed like it was inevitable because they had enough time to push through the changes and write the constitutional law before the next general election. And then out of the blue the emperor announced, "I want to retire. Please implement the constitutional changes that would allow me to do so". These changes, perhaps not incidentally, meant that Abe could not draft his changes until after the next general election.
It has been suggested (by the NHK no less -- unfortunately you'll have to take my word for it, but that's where I heard of the notion) that this was an intentional move by the emperor. Who knows... However it is true that he has been suffering from poor health and has not been able to perform all of his duties.
Anyway, the general election was held and Abe and the LDP survived, so presumably they will go ahead with their plans. However, the latest I've seen is that there is still considerable opposition in the populace (with over 50% of the population opposing any changes to the constitution). The last I heard Abe has said that he might not go forward with it.
They are under increasing pressure from the US, though. Also, Japan has wanted the US out of Okinawa for a long time, with the US always promising that they will leave as soon as their base in Guam is complete (which never seems to happen). I don't think the US will ever leave Okinawa without Japan being able to take military action the in the sea of Japan because otherwise China will pretty much have free reign there. At the moment, all Japan can do is show up with the coast guard and say "Please leave. Pretty please".