If I sell my inherited house in India, how do the proceeds get moved to the US and taxed?

Sale proceeds of Indian property go into your NRO account. To repatriate them to the US you use the USD 1 million per financial year NRO limit and file Forms 15CA and 15CB certifying Indian taxes were paid. India taxes the capital gain, and the buyer must deduct TDS on the sale (a high rate for NRI sellers unless you obtain a lower-deduction certificate from the Indian tax office in advance, which is worth doing). On the US side you also report the gain, using your cost basis; because you inherited it, your India cost basis rules and your US basis (generally fair market value at the decedent's death) can differ, so track both. Claim a US foreign tax credit for Indian capital-gains tax paid. Consider a CA and a US preparer who both handle cross-border cases.