How does the Republican Party thrive on rising income inequality? Answer: The more financially squeezed the middle class feels, the more susceptible they are to the Republican promise of lower taxes.
This proportion benefits Republicans at election time; but it squeezes the middle class all the more. We cannot satisfy aspirations of a democratic society committed to equal opportunity in contemporary times without considerable government spending (investment) to foster a more level playing field. When Republicans succeed in crippling government with their lower taxes message, they tilt the playing field to favor the wealthy even more and the middle class falls further behind. The success of their message since the advent of the Reagan administration is a major cause of the rising income inequality that troubles us now.
A simple example: Consider Social Security, one of the most successful anti-poverty measures by government in U.S. history. Republicans commonly preach that Social Security is heading toward bankruptcy; but they refuse to raise the cap on income subject to Social Security taxes ($117,000 a year currently). They would rather push the system toward bankruptcy than raise taxes on higher income persons. If they succeed, the wealthy will not suffer; it's the less than wealthy who will bear the brunt of their anti-tax policies.
I could cite numerous other examples. But the basic point will remain the same: As long as the middle class buys into the Republican pledge of lower taxes and less government, they will have fewer and fewer opportunities to maintain themselves as middle class.
While kicking yourself in the teeth is nearly impossible, you can achieve the mental equivalent by simply falling for Republican rhetoric.