Analog circuitry for measuring capacitive touch
So, is there an analog circuit which can achieve this without any microcontroller?

Sure. The classical way of measuring capacitance is building an oscillator whose frequency depends on the capacitance.
From there, multiple ways:

if you waveform is already stable in amplitude, i.e. the amplitude doesn't depend on any factors and especially not the capacitance, a simple low-pass filter can be used to dampen the wave proportionally to its frequency; a rectifier following that (and another, but higher-frequency low pass, to get rid of the harmonics) can convert that then into an inversely frequency-proportional voltage.
If you can't make reliable statements on the oscillation's amplitude, a number of ways, easiest probably being strong amplification sufficient for converting it into a square wave, can be walked to get a constant-envelope waveform.
If you have complexity to spare: use a reference oscillator, and a phase error detector as in a PLL. The error signal's average is proportional to the frequency deviation.

Other ways include delta-sigma conversion of the capacitance through measuring the time it takes to charge it; but now we're getting dangerously close to digital domain again, and you wanted to do it in analog.