In its current form, no. But that's not to say another competitor wouldn't have stepped up.
Sorry for the late reply. I had this post halfway finished the other day but I've been preoccupied with other stuff for the past few days.
I have my doubts that someone else could have succeeded had Nintendo never entered the console market. The only two possibilities would have been the Atari 7800 and the Sega 8-bit line (SG-1000, Mark-III, & SMS)
The U.S. market was effectively DOA in 1985. The only survivor was Atari, and even then only the brand name (the original company having been dissolved and its assets sold off). The Atari 7800 was released in 1986 when the crash was still fresh in people's memories, and the Atari name could very well have carried a stigma with it, consumers wary of a repeat of the events that had transpired just a couple of years earlier. Furthermore, the Atari 7800's library didn't have anything that really pushed the medium forward, and consisted mostly of ports of arcade games, many of which were present on the 2600. The 7800 was literally a more advanced 2600, and Atari wouldn't have been able to convince many people otherwise. Put short, the Atari name may have been toxic at the time, and Atari's games probably wouldn't have sold people on the 7800 like Nintendo's games did for the NES.
As for the Sega 8-bit line, the SG-1000 struggled mightily in Japan. It released the same day as the Famicom, but failed to make the same impact. Why? As far as I'm aware, NoJ didn't have the same burdensome licensing policies NoA would have later in the decade, probably because Nintendo didn't have the 10NES lockout chip at the time so there weren't really any obstacles to third-party development on Sega's system. The Famicom arguably succeeded on the merits of its own games, most of which were first-party back then (though Namco and Hudson Soft were supporting the FC back in '83-'84). The SG-1000 meanwhile didn't really have any notable games, and while Sega did have some arcade games in the early 80s, few of them were notable, either, at least nothing to the level of, say, Donkey Kong (which was a launch title for the FC). Could the SG-1000 have succeeded in Japan without any big killer apps?
Moving further along, the Mark III/Master System could have fared a bit better, especially with it being essentially an upgraded SG-1000 that was more powerful than even the NES, but again, there's the issue of games. Nintendo already had a strong library of games by 1986, most notably Super Mario Bros., which was one of the most groundbreaking titles in history. SMB is arguably the biggest factor that contributed to the NES being a commercial success in North America, and I think that even without NoA's strict licensing practices (which were probably justified given the factors that led to the Crash of '83) the NES still would have dominated. Would Alex Kidd, Wonder Boy, Hang On, Out Run, Fantasy Zone, or Space Harrier had the same impact as SMB did in a world where the NES never existed? Could they have gotten enough people sufficiently excited so as to revitalize the console industry? And let's not forget that the Mark III continued Sega's failure to generate interest in Japan, showing that Japanese audiences simply didn't care for Sega's console games.
As for third parties, there's no guarantee that they would have developed for the SMS like they did for the NES. The Famicom already had better third-party support than the SG-1000, and again that's without any strict licensing practices. In fact, I seem to recall that Sega didn't allow third parties to even publish on the Mark III/SMS until like 1988, so Sega was probably just as unwilling to cultivate third-party support as they were unable in the West. It's entirely possible that in a world without an NES, companies like Konami, Capcom, Square, and Enix would have kept making games for PCs and home computers (and arcades, for the first two) and ignored consoles entirely.
Sega also didn't have the marketing chops that Nintendo did back in the 80s. NoA's ad people arguably did just as much to get people interested in the NES as the games did. "Now You're Playing With Power" is a phrase forever etched in the minds of those who grew up with the NES, and is probably well-known among many younger gamers who were born after the NES was a hot commodity. Meanwhile, Sega's marketing department didn't really make a serious effort until the Genesis came out. While the Genesis struggled early on thanks to Nintendo still having a de facto monopoly, it wasn't for lack of trying (see "Genesis Does What Nintendon't").
TL;DR, Nintendo was uniquely positioned to revitalize the console market in the U.S. and establish it in Japan. They had the games and the marketing needed to generate large amounts of interest in console gaming in one market where consoles were effectively dead and one where consoles had yet to take root. I do not think Atari or Sega could have replicated what Nintendo did. There wasn't simply a vacuum waiting to be filled, because the demand for consoles didn't exist in the Japan of 1983 or the America of 1985. Consumers had to be made to get excited for console again, and I think Nintendo were the only ones that could have done that. Without the NES, the console market would likely have remained dead or limping along as a niche market at best for the remainder of the 80s. And who knows if the console market might have been revitalized sometime later? The mid 80s may have been a pivotal time period that could have determined the viability of consoles as a product, and without that NES-led revival, we might be playing everything on PC and/or arcades.