I mean you already see some of that, as any household with a household income >$250K pays the same in taxes. 250K is way different than $1 million. For me it would depend on what the flat percentage was. Everyone has to pay taxes, whether you like it or not. So if the flat tax rate is not significantly different from the tax rate that lower income are already paying then what would be the difference?
Your scenario also doesn't take wealth distribution into account. In the US the 1% are terrible with sharing their wealth with the rest of the 99%. That's not the case in the OP. He's sharing his income and also improving her standard of living.
It is a simple question. He is making double what she is, is it "fair" they pay the same rate?
Some could argue the rich provide jobs and the larger nominal value of their taxes improves all our lives to a greater degree than my measly middle income wages can. This nominal value will be higher even if they pay the same rate.
/I know in practice, because of tax avoidance strategies, the rich end up paying a lower rate. But in theory and law, we have progressive taxes to distribute wealth fairly.