> "O, swear not by the moon, the inconstant moon,
> That monthly changes in her circled orb,
> Lest that thy love prove likewise variable."
Romeo is not saying the moon is pale and sick with grief. He is, however, making a point about the moon's fickleness: its constant waxing and waning. He uses this to draw a parallel to Juliet's possible fickleness in love. He fears that her love for him might be as fleeting and unreliable as the moon's phases.
The phrase "pale" is used to describe the moon's color, not as a sign of sickness or grief. It's a common way to describe the moon's appearance, especially at night.
It's important to remember that in Shakespeare's time, the moon was often associated with fickleness and unpredictability, making it a fitting image for Romeo's anxieties about Juliet's love.