No. Users with public accounts do not have the permission to publish hosted feature services to ArcGIS Online. To publish services, users must have an organizational account and hold either a Publisher, Facilitator or Administrator role.
Users with public accounts have the option to open a free 30-day trial organizational account subscription if they want to publish hosted services. However, when the trial organizational subscription period ends, if the user converts the account into an ArcGIS Online public account instead of purchasing a full organizational subscription, any hosted feature services that were published during the trial period are lost.
Note: Users can still upload data to their ArcGIS Online public account by exporting the feature classes as a shapefile and zipping it using Windows Explorer. However, only 1,000 features can be uploaded.