One thing is for sure, housing prices would skyrocket!
Sure, in the hypothetical of instantaneously doubling the population without any time for the market to adjust for increased demand. But if the population doubles over the course of 30 years? That's only about 11 million people a year, the market would expect and adjust to the influx of people easily. Governments would also ideally be devising initiatives and changing policy to promote affordable housing. The high cost of housing in the US is its own issue anyway.