Well, the reality is that the Soviet Navy likely wouldn’t have expanded all that much during the 1990s. Even without the actual downfall of the Soviet Union, it was not doing well economically. It had been on a downward trend since Brezhnev really. And, besides relationships were warming under Gorbachev. Germany reunified in 1989 and START was signed in 1991. Sure, that was maybe because Soviet Union was dying but it also had to do with the nature of Gorbachev himself.
That being said, the Ulyanovsk likely would’ve been completed for the sake of principle and as a PR piece, not unlike Kuznetsov. However, despite the questionable tactical relevance of the Soviet Carrier aside, the US would likely produce an extra 5 Nimitz to maintain the 15-carrier requirement, but it’s doubtful we’d expand beyond that. Alternatively, the GRF-class would’ve been accelerated to start replacing ships sooner. USS America would’ve certainly undergone SLEP instead of being decommissioned and sunk. Enterprise would’ve been held onto for a tad bit longer, or replaced by a Nimitz. In fact, most of the conventional carriers would’ve had their retirement dates adjusted. I have a GAO report laying around somewhere from ~1987 that outlined the approximate replacement/SLEP schedule of the CV fleet that I should find.
The Des Moines were not making a come back for the same reason they weren’t tapped in the 1980s- they lacked the reserve buoyancy necessary to install modern systems. Besides, the Newport News had its center gun of turret #2 blow up in ‘72 off Vietnam. Add to that Iowa’s turret explosion in ‘89, I doubt the Navy was in a rush to reactivate more ships firing 60-70y/o munitions. Considering they were just a (relatively) cheap stop gap Missile barge with excellent propaganda value, they weren’t needed long term. By 2008 at the absolute latest, I’d say the last of the Iowa’s are mothballed or decommissioned. If the spare parts even lasted that long. That would also probably extend to the NTU ships and nuclear cruisers by 2008-2010 aswell.
As far as Submarines go- the Seawolf-class would’ve continued production and would likely still be built to this today. Considering the multi-year gap in production, the we’d likely have a good ~30-35 Seawolfs by now. Keep in mind the USN is already pumping them out as fast as they can, the SSN fleet is arguably the most in demand assets in the Navy, and they’re running their boats to the breaking point to reach mission requirements.
However, the Ohio-class would remain 100% unchanged unless START didn’t happen.
The big “What if?” in this scenario is if 9/11 happens. Even in this alt-history universe, that would have a profound impact on the state of the world. Hard to say if Soviet Union would outright support Al Qaeda like we supported Mujahideen just 15 years prior. I’d lean towards no, but who knows. But considering the Soviet Union would still be a threat, I don’t think Bush would’ve commit to Iraq.
Speaking of Bush and Iraq, let’s talk about his awful VP. Realistically, Grumman will still be chugging along. I don’t know why Cheney had such a hate boner for Grumman, but he would not be able to cut the F-14 as early as he did while the USSR is around. I think by now the F-14 would likely be retired, but far more recently then 2005. And instead of being replaced by ugly terrible no good bloated “super” Hornets*, greater emphasis would’ve been placed on the NATF & A-12. Especially considering the construction of the Ulyanovsk.
*I hate Hornets like Cheney hates Tomcats.
I also doubt the LCS program would exist. It was drafted during a time when the US Navy didn’t really know who it’s enemy was, and it reflects that. If the USSR never disappeared, I’d say they’d probably order a more conventional frigate design.
Overall, would we ever achieve 600 ships? Maybe briefly, but still doubtful. Unless the Soviet Union and US were in a old fashioned Naval arms race, which the USSR couldn’t afford in the 90s, the US just doesn’t currently have the military shipyards necessary to pump out ships faster then we are retiring them. And the older ships, while sure pad the fleet number, will just come to hog the budget while adding little actual value.