What is the effect of a varying Vcc on the output pin of an IC?
First let's put it on the table that the Vcc pins of IC chips really need to have local bypass capacitors to GND or the return  pin of the IC for those that may use a different name such as Vss.

If the Vcc of the IC varies it will generally have a direct impact on the high level output level of a digital IC. To a lesser degree it will also have an effect on how well the output pin can drive to a low level due to fact that the bias to the pulling down transistor comes from the Vcc supply.

For linear circuits, lets use op-amps for this discussion, the output voltage is generally not operating at the rails (if it was it would act more like the digital chip) and is set by feedback resistors and the nature of the input signal and biasing. For these applications you would really hope that variation of the supply voltage, within reasonable limits, would have no impact on the output voltage. In real life it will have some impact and so to quantify this effect the data sheet of a part will specify a "power supply rejection ratio" that specified how much the output is affected.