Does the US government officially recognize the islands of Taiwan to be part of the People's Republic of China?
Apparently not, at least since 1971.
AFAIK, the USA does not formally recognise ROC independence. I think the USA has more formal relations with the ROC now, but I am not sure it has been officially recognised as an independent country. I think there are two main obstacles - the big one is that the PRC views Taiwan as a rebellious province. The other is that the ROC (or since the brutal military dictatorship ended in 1987, some in Taiwan) considers the PRC to be illegitimate and views itself as the 'real' China.
We all hear a lot from the pearl-clutchers on this forum about the PRC's 'unreasonable' claims to the South China Sea, but no one really makes much of an issue about how the ROC maintains the EXACT SAME claims. The view that the South China Sea is Chinese is a Chinese thing - not a 'red China' thing.