The middle class does pay taxes; in fact, they probably get the short end of the stick. You say that the "majority" does not pay "any" (we'll assume by "any" you mean "federal income," since that is the only way your assertion is accurate) taxes, but what you don't acknowledge is that the line at which that occurs is drawn at around $45,000 per year of income per household. Most people would say that households which are making less than $45,000 per year in income are not middle class households. So your point that "the middle class needs to start kicking in their fair share" is clearly based on an incorrect premise. You will need to reconsider your conclusion.
Second, the problem that you see is actually solved very easily. If those who pay the largest share of federal income taxes (top 50% of income earners) wish to avoid this burden, they can do so easily by giving their income to the bottom 50%. You see, all people are treated the same by the tax code. It is only income that is treated differently. If you do not like the fact that only 50% of people pay income taxes, the problem can be solved by distributing more income to the bottom 50%. In other words, what you are pointing out is not a tax problem, it is an income distribution problem. The poor would gladly pay more in taxes if they had more income. And if the rich do not want the tax burden they have, they can easily absolve themselves of the burden by giving their income to the poor. We don't see that, because it is more of a burden to have low income and pay no income taxes than it is to have a high income and pay income taxes. There aren't a lot of high income earners begging working class people to trade places with them, are there?