Only 1% of the US civilian labor force earns the minimum wage. So you're talking about a small fraction of the total US pay base. It wouldn't even remotely dent the 77% number, first because the minimum wage affects so relatively few, and second because the wages in the group are so small it won't meaningfully impact the median income that primarily drives the 77%.
To put it into perspective, by your 63% number, it comes out to about 1 million women versus 750,000 men. Shifting the pay of those 250,000 minimum wage workers (that represent the sex gap on minimum wage) by $2 or $3 probably will not even move the 77% number to 77.25%.
Close, but I don't think that quite accounts for the whole story. For instance, if minimum wage is raised, it depends on how much it's raised by to determine what % of workers will have increased wages. If we took it up to (purely as an example) $15/hr, then everybody earning less than that would also see a pay raise, but would not be included in the 1%. I don't know where you got your previously cited statistics, so I can't exactly run the numbers to see the impact, but I imagine it could be a sizable effect.
It would increase it for the wrong reason, and only in the short-term, you pre-suppose that increasing the minimum wage won't lead to other wage increases. Why would someone who is getting paid more than minimum wage to perform a job which requires a skill be ok with being paid less than they were before, in relative means?
The increase in minimum wage does cause increases in near-minimum wages as well, but as you go up the salary scale (which is stretched quite longly, thank you very much), the impact of minimum wage increases is reduced, leading to an overall reduction in inequality.
I'm sorry but there will be no reduction in the overall income gap with a raise in minimum wage. You admit the minimum wage increase will cause the jobs near that rate to increase as well as the medium income levels. The high end of the income scale is so far "stretched" as you say and held by so few people that the effect of the minimum wage increase is insignificant. Infinity -1 is still infinity.
Of course there will be the resultant unintended consequences such as the price of housing rising due to more people having a little bit more to spend, seeing as housing and healthcare are both increasing much faster than incomes and inflation it is likely these two will see the windfall and end up resulting in the lower classes having even less disposable income.
http://www.theatlantic.com/politics/archive/2015/04/who-are-...
Increasing minimum wage would effectively increase the 77% number also.