A recent study from Israel suggests the answer is, “No.” A preprint of the report–“Protection of previous SARS-CoV-2 infection is similar to that of BNT162b2 vaccine protection”–was posted on 4/24. From the Abstract:

Understanding the level of protection of previous infection compared to that of vaccination is critical for policy making. We analyze an updated individual-level database of the entire population of Israel to assess the protection efficacy of both prior infection and vaccination in preventing subsequent SARS-CoV-2 infection, hospitalization with COVID-19, severe disease, and death due to COVID-19. Vaccination was highly effective with overall estimated efficacy for documented infection of 92·8%; hospitalization 94·2%; severe illness 94·4%; and death 93·7%). Similarly, the overall estimated level of protection from prior SARS-CoV-2 infection for documented infection is 94·8%; hospitalization 94·1%; and severe illness 96·4%. Our results question the need to vaccinate previously-infected individuals. [Statistical range data omitted.]