Does the US Government shutdown make me stateless (like Viktor Navorski)?

The definition of Statelessness is:

Statelessness is a legal concept describing the lack of any nationality. It denotes the absence of a recognized link between an individual and any state.

A de jure stateless person is someone who is "not considered as a national by any state under the operation of its law".

A de facto stateless person is someone who is outside the country of his or her nationality and is unable or, for valid reasons, unwilling to avail himself or herself of the protection of that country. This can be a result of persecution or a consequence of lack of diplomatic relations between the state of nationality and the state of residence.

Your state (the USA) still exists. You also still have a passport that proves that link. On top of that, your primary place of contact, the US embassy in India, also continues to operate. Therefore there is no risk for you to be stateless.

Victor Navorski is a fictional person. I would rather refer to the person his story was inspired by, Karimi Nasseri. Mr Nasseri lost his link to his home country, since it's passport was lost.

I would be highly surprised if US passports for ca. 6.3 Million Americans living abroad become invalid. The worst case would be of losing a passport when you are required to have one (for example a customs agent checks you) AND the embassies would be closed for a longer period at the same time to confirm your citizenship. However, even in this case, given the international status of the USA, I would assume that the majority of countries world-wide would not go on a hard-line mode and detain all US citizens living in their country immediately, given the amount of people affected. The only countries I could imagine giving you trouble would be North Korea, Iran etc, but those do not have a US Embassy to begin with.