"The outdoor electrical box has come away from the exterior wall."

A lot depends on what the wall is made of.

If you turn off the power to the circuit you can take off the front of the box, then remove the receptacle and see how it is/was attached to the wall.

Sometimes all that is needed are longer screws to get a better bite into the wall material.

Masonry walls are a little harder to deal with.

The "The" indicates that this is your main service panel. Is that the case?

It all depends on what The outdoor electrical box is.

If "The outdoor electrical box" is either your meter box, main disconnect or your main breaker panel (with integral main disconnect), you should have a licensed electrician perform this task. Removing the meter would likely be required in order to de-energize the box and you must be licensed to do so.
If it's an outlet, switch or junction box, turning off the appropriate circuit breaker(s) would be sufficient.