No.Don't forget watts. Big speakers pull more watts.
Big speakers can survive more Watts.
An 8 ohm tiny speaker and an 8 ohm huge speaker both use exactly the same amount of power from an amplifier. The amplifier provides a voltage signal. The amount of power "pulled" is the voltage squared divided by the speaker impedance. Speaker size is not in the equation.
The huge speaker will probably produce more sound level from the same amount of amplifier power that feeds the tiny speaker because a big speaker is more sensitive due to its big cone moving more air than the tiny speaker.