That's not what science actually says about the expansion of the Universe. Nasa talks complete nonsense here.
Gravitationally bound objects like galaxies and galaxy clusters do not increase in size along with the Universe, much less smaller things like stars and planets. Once objects are bound together by gravity or by another force, the distance between them does not further increase. Even the so-called dark energy or cosmological constant only acts as the tiniest repulsive force between objects, but that force depends only on the distance between them, and as such, does not grow with time.
The expansion of the Universe only becomes relevant outside of the Local Group of galaxies, inside it it has absolutely no effect. Our Milky Way will remain bound together until the last stars burn out, trillions of years into the future.