He needn't have worried. Galadriel certainly had sense enough to wear to shoes in the snow. And after having crossed the Helcaraxë, this cold was nothing. She withstood some of the harshest environments in Middle-Earth; a little snow would do her no harm.

"You forget that I am a healer. I know better." Her expression lightens slightly as he kisses her hand. As soon as he releases it, she reaches over to massage his shoulder.

"I am not so fragile." He didn't need to protect her- from worry or from the cold.