No that's not true. The federal government also collects capital gains, corporate taxes, FICA, estate, gift and payroll taxes. FICA and payroll taxes are regressive. Capital gains is in reality regressive because it is a much, much, lower rate than the rate they tax wages at and the rich get a much larger percentage of their income through investment than other people do. Estate taxes are hard to classify. They only kick in after $2.5 million and they are a type of income that mostly is collected by the rich, so it is mostly a big fat tax exemption for the rich on a source of income. But, they do kick in above that again now that we let that part of the Bush tax handouts expire. So, I wouldn't really say it fits neatly in either regressive or progressive territory. Same with gifts. So 3 of the 4 federal taxes are regressive, but you just want to look at the 1 out of 4 that is progressive and draw conclusions based just on that?
Not to even mention state taxes, which are much more regressive. Sales taxes are steeply regressive. Property, unemployment and disability taxes are all regressive. And most states either have no income tax, a flat income tax, or a marginally progressive income tax with maybe just two brackets, so they aren't even offsetting the regresiveness of their other taxes like the fed is trying to do. So, even just looking at all federal taxes is misleading, since the overall tax burden people face is much more regressive than just the federal part.
So, statements like you made about "how can the Democrats say the rich aren't paying their fair share" that only cite to federal income taxes are at best misleading. Whether or not they're paying their fair share would depend on the percentage of ALL taxes that they pay- state, federal, income, sales, property, capital gains, FICA, etc.