But whether I can arrive at the formula of the output capacitor
without any formula involving the switching frequency?
The minimum value of output capacitance is based on the amount of ripple voltage; more capacitance means lower ripple voltage; higher switching frequency (same capacitance) also means lower ripple voltage.
The ripple voltage is directly related to the switching frequency so no, I would say you need to know the switching frequency to predict the ripple voltage based on values for the inductor and capacitor.
The inductor and capacitor form a low-pass filter and, that low-pass filter frequency has to be significantly lower than the switching frequency or you'll get large ripple artefacts. All normal (commonly used) buck converters are designed to reduce ripple significantly.
Hence, both the inductor's value and the capacitors value AND the switching frequency dictate ripple voltage on the output. Imagine an LC low pass filter and you apply a square wave to the input; if the cut-off frequency is significantly lower than the fundamental frequency of the square wave (aka the waveform from the switching transistor) then the output ripple is low.