Creatures can't be conscious at zero HP
Spare the dying merely stabilises the target. It can't restore consciousness because it doesn't restore hit points. This is because:
- Consciousness requires at least one hit point1.
- (Conversely) stable creatures remain unconscious until they gain at least one hit point.
- Spare the dying stabilises a creature, but doesn't restore any hit points.
Consciousness requires at least one hit point
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious... This unconsciousness ends if you regain any hit points.
Basic Rules, p.79
Stable creatures remain unconscious until they gain HP
A stable creature doesn’t make death saving throws,
even though it has 0 hit points, but it does remain unconscious.
Spare the Dying merely stabilises creatures
As per the spell's description:
You touch a living creature that has 0 hit points. The creature becomes stable.
The creature becomes stable, but does not regain any hit points and therefore remains unconscious. As such, it is unable to move, speak or take actions:
An unconscious creature is incapacitated (see the condition), can’t move or speak, and is unaware of its surroundings.
Basic rules, p.172
1 There are a handful of specific game effects that override this general rule, such as the Zealot Barbarians' Rage Beyond Death feature.