You're right, it isn't socialism. Socialism doesn't mean the government doing things. Capitalism and socialism are totally mutually exclusive. There's a sudden tendency to equate welfare states with socialism. This is totally wrong and a coopting of the word socialism.

Socialism is achieved when the workers own the means of production, i.e. workplaces are democratically controlled. This has yet to occur on a large scale anywhere on the earth. Communism is generally agreed to be "after" socialism where all money, states, and such are abolished.

It isn't a spectrum where you can be "somewhat capitalist, somewhat socialist." A society is either capitalist or socialist/communist. Western European nations are something called "social democracies." This means a welfare state has been formed which makes capitalism a tad nicer for the working class. The fundamentals of capitalism (private ownership over the means of production) has NOT been replaced by socialism (worker/common ownership over the means of production.)

A public health care system is just a government program. It isn't socialist. This is a massive misunderstanding which needs to be addressed. At least we can talk about socialism without all the stigma, I guess. r/socialism and r/socialism_101 can answer further questions and both have excellent resources available.

TL;DR: No, single payer healthcare isn't socialist. Furthermore, there is no scale that goes from no government control (capitalism) to mixed control (socialism) to total control (communism). The correct sequence is capitalism (private control over means of production (capitalism) -> socialism/communism (worker control over means of production).