Siamese (conjoined) twins wherein one twin becomes ill with an illness yet the other remains free of the illness, despite sharing the same blood supply is the clearest of empirical falsifications of the the claim of contagion.
Refer to these documents on Masha and Dasha Krivoshlyopova, Siamese twins born in 1950 and extensively studied by Russian health authorities. One twin would become ill, have fever, pneumonia, or measles and the other would not have any illness, or symptoms.
This is an empirical falsification of contagion in general, and viral contagion in particular.