The US already spends a large percentage of GDP on public health care (medicare and medicaid). It's just that it spends much more in private on top of that.

I quickly found some numbers from 2008. All numbers are a percentage of GDP:

US: 7.4% public, 8.5% private

Switzerland: 6.3% public, 4.4% private

Sweden: 7.7% public, 1.7% private

France: 8.7% public, 2.5% private

graphic:

https://kaiserfamilyfoundation.files.wordpress.com/2011/04/h...

kff study:

https://www.kff.org/health-costs/issue-brief/snapshots-healt...