Self-resonance frequency of MLCC in Buck converter circuit
I think, there should not be instability issues as long as the resonance frequency of the inductor/capacitor combination is well above or below the 1MHz.


Stability depends mainly on the behavior of the circuit within the bandwidth of the control loop and slightly above. As another answer says, a key parameter is the phase margin of the control loop. This is measured at the frequency where the open loop gain passes through unity. 

This frequency is generally much lower than the switching frequency of the regulator, typically by as much as 10x, so about 100 kHz in your example (but read your datasheet and analyze your design to figure out what it is in your particular circuit).

Edit: I should add, the output capacitor is also important to smooth the ripple from the switching waveform. This is a separate issue from the stability of the control loop. As another answer says, the switching waveform will contain harmonics far above the fundamental frequency, so you will likely want to include some lower value, higher SRF, capacitors in parallel with your 47 uF MLCC to deal with those components.