Let's wind back the clock 23 years. The 6th generation of consoles is underway. The PS2 is off to a flying start, and the Gamecube has just released. 

In this timeline, however, something has changed; Microsoft decides not to the join the console market. Xbox never happens.

How do you think the following years and generations up to the present day play out in this scenario; what would gaming look like without Xbox?

Well it turns into a Sony and Nintendo race where Nintendo keeps rare. Which is....an interesting debate. It could be some better off but still smoked Gamecube leading to a Wii with Rare games while Sony would probably have even less reason to be humble than in the PS3 launch. Which could have some implications.

...Unless this leads to a bolstered Dreamcast that somehow survived and ends up creating some 'Neptune' console with or without a ring of death.

I thought Nintendo sold their Rare stock, which would cause Rare to be in a situation where it needed to be acquired (keep in mind Nintendo rejected acquisition)?

Sega would still have no reason to drag Dreamcast out more, or at the very least not make another console.
Rare goes multiplatform, as I'm skeptical of Sony buying them.
PlayStation Network is the torchbearer for console online multiplayer and functionality. No way is Nintendo going to do it instead of Xbox Live.
Sony is even more arrogant developing the PS3, as there is no Xbox 360 to compete with it.
Nintendo doesn't change much, at least not off the top of my head. GameCube will do better, but I'm pretty sure Xbox stole more PS2 sales than GameCube sales.

Some interesting questions arise:

1) Would Sony have dropped the price of PS3 without the X360 competition?
2) Would cross-play be a thing, considering how much Sony was opposed to the idea?
3) Would Sony have implemented backwards compatibily after the PS3?
4) How much would the PS4 have cost?
4) How much would the PS5 have cost?
5) Would Sony subscription service exist as it is if GP never existed?

And I don't think much would have changed on Nintendo's side.

Honestly no, no, possible, a lot, and a lot. 

Honestly I'm kind of curious if any other company but Microsoft could come in to fill the Xbox role? Because otherwise....well it could get very funky.

Biggest thing I care about is Halo would’ve either been a Mac exclusive or Bungie would’ve shutdown and Halo wouldn’t have ever happened.

Also Bethesda’s entrance into console gaming would’ve take longer since they didn’t even know if Elder Scrolls would’ve worked on console.

Nintendo had 49% and Stamper brothers 51%. Microsoft bought the 51% share and Nintendo sold it's 49% share to MS afterwards. It could be that Microsoft offered Stampers more money than what Nintendo considered Rare being worth and didn't buy the stock even if they could have.

I don't think outside of MSX Microsoft have or have had much relevance to gaming, Xbox even less. Gamecube had done better in 6th gen and PS3 had done better in 7th if Xbox hadn't existed. However, I think at least some of the last decade's entrants would have had at least moderate success without Xbox.

We would be one extra generation behind on online gaming.