I have my doubts. Over the last 50 years the ratio of US per capita health care costs and European per capita health care costs (or per capita health care costs for most of the rest of first world countries) has stayed about the same.
In other words, health care costs have risen at about the same rate throughout the first world over the last 50 years. So if in 1970 the US was paying say 3x per capita what some other country paid the US would still now be paying about 3x what they are now paying. Both would be paying maybe 35x now than they were in 1970.
Over that same time both US and European obesity rates went up, but they went up way more in the US. If obesity was a major factor driving health care costs then I'd expect US health care costs to be rising significantly faster than European health care costs.