Defeat every other gym. The Viridian City gym is actually your last gym, not your first.
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The gym in Viridian is actually the last gym. You'll unlock it after getting all the other badges. You should continue to Pewter through Viridian Forest.
You don't until you beat the 7 other gyms. After that fly back to Viridian city and the leader will have returned
The eighth gym is back in Viridian City, where you have already been, but it won't unlock until you have all of the rest of the badges. Once you defeat Blaine at the Cinnabar Gym, you can take Route 21 back to Pallet town and either continue to Viridian, or fly there. The gym leader is someone you have met before.
All you have to do is get to Viridian City.
It is in Viridian City. After you have beten the 7th gym leader it is open