Finding a Hamilton path in a Complete Euclidean Graph is in P
This assertion is, in fact, true. A complete graph (which doesn't even have to be Euclidean) always has a Hamiltonian path: number the vertices in an arbitrary order, start at the first one, then proceed in order to the last (and then back to the first to make a cycle if you want). Since the graph is complete, the edges you need will always exist.

Thus, the problem can be decided in $O(1)$, which is polynomial, so it is in $P$. The algorithm is trivial:

define doesThisCompleteGraphHaveAHamiltonianPath(G):
    return Yes