Would he approve of all these spinoffs?

Voyager? I think you mean Discovery. Rodenberry would never approve of Scum Drek Discovery, Picard and SNW.

Some of the spin-offs beginning with "Deep Space Nine" are dark and dreary so he wouldn't approve since they're far from his vision. They also don't address the social commentary like his two shows did. I haven't watched Strange New Worlds yet so I'm not sure if that has his vision.

DS9 - no (too dark, and set on a space station?)
VOY - maybe (as its a 'star trek', although not set on an Enterprise just some random ship doing 'Lost In Space' so maybe not)
Ent - maybe (unless he'd have thought it fcuked with established continuity)
JJ films - sort of (he was interested in doing a ToS prequel movie when he was making the original series; although would've probably frowned on the non stop action plotlines)
Disco - no way (hed be raging)
SNW - yes (as its basically 'Star Trek')
Picard - yes (as its a continuation of his TNG characters esp s3)

I don't care if Roddenberry would approve of the Star Trek spin-offs or not, he was TOXIC to Star Trek, his own creation.

TOS pilot: He creates that stupid boring cerebral storyline with Captain Pike and a female Number One and the Talosian aliens, something out of the 1950s stuff, and is a failure, so instead, a more action-packed pilot storyline is created, and so is the Enterprise crew we all know and love. This pilot is a success.

As far as I know, with TOS, he left the writing of episodes up to seasoned sci-fi veteran writers who knew what he was aiming for, and the show was great in the first season, and maybe the second. But something obviously went wrong by S3 because it turned to shit and was cancelled. I don't know for certain if Roddenberry had anything to do with the decline, but I would not be surprised if he did.

The Motion Picture is developed under his supervision in the 1970s, and it turns out to be boring, 2001-like but without the class of Kubrick, and is just generally unliked. Roddenberry was responsible for the "cerebral" vision, and thus was kicked upstairs out of the way of the creative process and a much better movie, Wrath of Khan is released that does big numbers at the box office and stabilises the Star Trek movie ship for future success.

The TNG pilot, Encounter at Farpoint, is made under Roddenberry's supervision, once again, and leads to an awful sanitised future with co-operation between all in the Federation and no conflict, and produces two of the worst Trek seasons ever, so bad that the show and franchise run is nearly cancelled a frickin' clip show episode! Then Roddenberry is kicked upstairs out of the way so that Rick Berman becomes creative lead, and along with other better people at the helm, proceeds to lead Star Trek into a bright and glorious future that lasts well until 2005 when that dickhead Les Moonves cancels the show because "he doesn't like it", but that's a tale for another time.

(continued)

So yes, because Roddenberry was removed from the helm of his own show, we ended up with the superior second 2/3rds of TNG, the excellent war-scenario DS9, and the TNG-lite show Voyager, because Star Trek has always been about exploration, but with a new twist, and the first prequel show, Enterprise, that started off weak but got stronger before cancellation. If Roddenberry had refused to leave, we wouldn't have these televisual gems to enjoy.

Abrams and Kurtzman, on the other hand...! Again, another story.