I would say no, and for several reasons.

One of the reasons college tuition is so unaffordable is government subsidies. Before the government guaranteed student loans, tuition was much lower. It was possible and common for people to hold down a part time job, live at home, and pay for college themselves. Colleges did not charge more than people could afford, because if they did they wouldn't have any students. A side benefit of this arrangement is that people would have not only a college education, but real life experience in a job. Now, many people go to high school for free, get a subsidized student loan and spend four more years as a student, and at 21 or 22 they graduate without ever having had a real, adult job. This makes them much less employable. Free higher education would only make this worse.

You might say the solution is for the government to set prices but this doesn't work. If prices are set by the government, this discourages innovation. European higher education is "free" for students (not taxpayers) but they still come to America because American universisites are widely acknowledged as the best in the world. What incentive is there for a university to hire the best professors, engage in quality research, and attract the best students if all universities get the same amount of money from the government?

The real solution is to end the government subsidies of student loans, which will allow tuition to return to affordable levels. Also, K-12 education needs to be fixed because in other countries they're doing stuff in 6th grade that high school seniors study here. Freshman year for many students is spent on remedial work. A high school diploma is essentially useless, and it's not due to money - it's due to lack of discipline and dumbed down standards.