Since your toe nail got ripped off there is a invreased chance of infection to your nail bed.So, I would suggest a topical antibiotic for application.
Yes you can leave the toe open provided if you dont go outside often/bare foot
Next Steps
Oin.T-Bact L/a Twice a day/SOS
Tab Hifenac-P 1-0-1 * 3 days
Tab Pan 40 1-0-0 before food *3 days
Cap.Amoxy 500mg 1-1-1 * 3 days
Health Tips
Take bath as if you take normally and wipe the wounded area after taking bath. Make sure no debris are formed in nail cleft periodically.
Consult further if symptoms persists.