The correct answer is $\boxed{\text{A}}$.
The power dissipation of a diode is given by the equation $P = VI$, where $V$ is the voltage across the diode and $I$ is the current through the diode. In this case, $V = 0.93\text{ V}$ and $I = 1\text{ A}$, so the power dissipation is $P = 0.93\text{ V} \times 1\text{ A} = 0.93\text{ W}$.
Option $\text{B}$ is incorrect because it is the voltage across the diode, not the power dissipation. Option $\text{C}$ is incorrect because it is a time, not a power. Option $\text{D}$ is incorrect because it is a power that is 10 times greater than the actual power dissipation. Option $\text{E}$ is correct because it is the only option that is a valid power.