With a simple resistive feedback that op amp will be stable for any frequency or gain with a phase-margin of 65 degrees. Typically you want at least 35-40 degrees. 1 MHz is just the gain-bandwidth-product, thus if the non-inverting gain is 2, the bandwidth would be about 500kHz.
You can check the stability by plotting the gain and phase of the circuit (Bode plot). Also look at the transient response of the circuit to a step or square-wave input. If the edges of the step output have significant overshoot or ringing then that indicates the circuit may be marginally stable.