Turn on a multi-functional pressure cooker (such as Instant Pot®) and select Saute function. Add oil and onion and cook until soft, about 3 minutes. Add garlic and cook until fragrant, about 1 minute. Stir in rice and cook until opaque, stirring frequently, about 2 minutes. Stir in 4 cups broth, tomatoes, and basil. Close and lock the lid. Select high pressure and set timer for 6 minutes. Allow about 10 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer’s instructions, about 5 minutes. Unlock and remove the lid.
Select Saute function and stir in remaining broth. Check rice for doneness; if needed, continue stirring until tender.
Stir in spinach and Parmesan cheese; cook until wilted, about 1 minute. Season with salt and pepper to taste.