No, fast food employees should not, all across the nation, be elevated to $15 a hour. There should not be a national minimum salary because the cost of living varies greatly across the states and from suburban to urban areas (an apartment in the suburbs of Houston is about 1/5 the cost of an apartment in New York City). Fifteen dollars an hour in say McAllen, TX would be a nice salary.
Often when people analyze this problem, they only account for the impact of raising the wage of the people currently at minimum wage. But you can't just do that. Obviously you have to also raise the wage of everyone that is below the new minimum wage, but to maintain the healthy ecosystem of employees being compensated for increasing responsibility and skill level, everyone in the organization has to get a raise.
Here's an example of a grocery store (that was my high school minimum wage job). Males were hired in as minimum wage bag boys and females were hired as cashiers at about $2 more. Employees that were good at the job could be promoted to stock person. Good stock persons could become department heads (produce, bakery etc.). Experienced department heads might become assistant managers and assistant managers might eventually become store managers.
We also have to consider the ripple effects not just within an organization, but across the employment field. If suddenly flipping burgers is worth $15 an hour, how much more will someone have to pay the kid that mows their lawn? How many people will start mowing their own lawn when it turns from a $50 a week expense into a $100 a week expense?
And finally, fast food jobs are not supposed to provide a living wage. They are intended to be part time jobs for students and retired people. If the system is broken and fast food jobs are all some people can find on which to survive, we need to address that breakage -- not put a bandaid on it by crippling the industry.
I don't think the people protesting expect to get $15. In negotiating there are three numbers. There is your MDO (Most Desired Outcome), your goal, and your LAA (Least Acceptable Agreement). $15 is their MDO. They'll be happy with $10.