Let me first correct a small misconception. Where you are talking about 'the magma ocean', you are implying that one exists. This is in fact false. There is no 'magma ocean' in the Earth at the moment (and it has been like that for several billions years). The lavas you see erupting in volcanoes are not coming from a magma ocean. They are coming from either the crust or the mantle, both of which are solid. Magma generation is a localised anomaly in what would otherwise be a solid mass.
That said, the upper mantle has a section called 'the asthenosphere' which is slightly more ductile than the rest. Think about chocolate in a hot day. Still solid - but it can flow given enough pressure. The modern lithospheric plates (of plate tectonics fame) sort of float on the asthenosphere already now, and that's why you have plate movement and earthquakes and everything.
The asthenosphere is quite deep - some 100 to 200 km deep below the surface. If a cosmic impact is going to create a crater 100 km deep in the Earth, no one will be left around to care about whether they will survive on the 'iceberg'-like lithospheric plates. Again - this is irrelevant. The asthenosphere is ductile in timescales of hunderds of years (at least). If you were to pick up a piece of it now, it will just be like any other rock (albeit quite warm).
A magma ocean event probably occurred very early in the Earth's history (which is not going to happen again since Earth is already here), and the moon forming event also likely liquified quite a lot of the Earth. But this is not going to happen since there is no other big enough body that seems like it's going to collide with us at the moment.