What about Rhaphiolepis?
Rhaphiolepis is a genus of about fifteen species of evergreen shrubs and small trees in the family Rosaceae, native to warm temperate and subtropical eastern and southeastern Asia, from southern Japan, southern Korea and southern China south to Thailand and Vietnam.
Originally I suspected Rhaphiolepis indica (Indian hawthorn), however this species produces pink flowers. After a little research I suspected Rhaphiolepis umbellata (Yeddo hawthorn) which produces white flowers, however they are scented.
I did find reference to a white flowering Rhaphiolepis indica - Rhaphiolepis Cosmic White - however the flowers appear to be too large.
I note @Marta (OP) confirms her location is Florida USA. Rhaphiolepis indica is popular in subtropical and tropical regions in Australia and I also read on Wikipedia that...
Indian Hawthorn is a mainstay horticultural specimen in southern
United States. It is often found in commercial as well as in private
Maybe @Marta (OP) could confirm whether the plant produces any fruit? I note that Rhaphiolepis sp. produce small fruit 1-2 cm (0.39–0.79 in) diameter and coloured dark purple to black.