How long can a healthy ecosystem remain stable?
Theoretically, forever. In practice, it depends.
A healthy ecosystem theoretically could remain stable forever because the biodiversity of a healthy ecosystem would make it resistant enough to most small changes that it would never collapse.
But ecosystems don't work in a vacuum, they're an open system. This means they're open to migration, natural disasters, disease, invasive species, and even climate changes (natural or unnatural). Geological records have suggested that some ecosystems have actually remained stable for hundreds of thousands of years.
Probably the most likely candidates for ecosystems that would last this long are old-growth forests because the trees, the sort of keystone for the ecosystem, last for thousands of years, deserts, which don't have as many factors for ecological disturbance as others (they are sensitive to disturbance, it just doesn't happen as often naturally), or any other kind of slowly-progressing ecosystem.