Combine water, sugar, and creme de cassis in a medium bowl. Stir until sugar granules are completely dissolved, 2 to 3 minutes.
Combine berries and lemon juice in a blender; process to a smooth puree.
Set a fine mesh strainer over a medium bowl. Pour small batches of the berry puree into the strainer, stirring vigorously to force the liquid into the bowl. Discard the seeds once most of the liquid has been removed.
Stir the berry puree into the sugar mixture until well combined. Cover with plastic wrap and freeze for about 4 hours.
Beat egg whites in a glass, metal, or ceramic bowl until until stiff peaks form. Remove the puree from the freezer and whisk until slushy. Fold egg whites into the puree until just blended. Cover and freeze until solid, about 2 hours.