Can I replace watch quartz with DS3231 32 kHz output?
Usually, yes - assuming you mean 32.768kHz when you say 32kHz, or the watch will run at the wrong rate.

First you need to know which IC pin is the input and which is the output, as Ignacio commented. A quick measurement with your oscilloscope will show a strong 32.768kHz signal at one pin on the crystal, and a weak signal at the other. The weaker signal is the input.

Now you need to connect the watch ground to the DS3231's ground, and capacitively couple the DS3231's 32.768kHz output into that input, removing the crystal of course. The correct value of capacitance is a guess, but 22-100pF is probably the range, and I would guess 33pf as a starting point. After the capacitance, you should see a signal approximately the same as the original. If it's much larger, reduce the capacitance and vice-versa. (You may have to remove the trimmer capacitance, or turn it to its minimum setting, i.e. maximise the input signal)

Oh and you still need the correct supply voltage of course - e.g. the original battery.

There should be a good clean copy of your 32.678kHz signal on the original (now unused) output pin.