Wasabi does support the ability to designate stored files as 'public' for access by websites and other methods. However, Wasabi does not currently support cross-origin resource sharing (CORS) that is often required for website use cases (see this AWS S3 CORS example for a typical use case). Wasabi also does not currently support the hosting of static websites using custom domains (see this AWS S3 custom domain example for a typical use case). Both of these features are under consideration for future development.
More info on Wasabi & CORS can be found here.