There were not 38 million Incas. Ethnic Incas numbered in the low tens of thousands. They had rapidly expanded their empire in only a couple hundred years or so from a relatively small area around Cuzco to a massive empire including millions of people of other ethnic groups and divided into four parts. So the Inca empire was a pretty recent development, and most of its inhabitants weren't Incas.

The diseases carried by the Spanish preceded the arrival of the Spaniards themselves. The Inca emperor Huayna Capac never met or saw any Spaniards but died of the smallpox they brought to the Americas. So when the Spaniards show up in the Inca empire, there was a civil war going on between Huascar and Atahualpa, two of Huayna Capac's sons.

At Cajamarca, the Spaniards manage to capture Atahualpa. Then they executed him.

It was all downhill from there. It's a really sad, complicated story. Kim MacQuarrie wrote a very interesting and easy to read book on the subject called The Last Days of the Incas.