Combine sugar, Amaretto, cherries, and blueberries in a medium bowl. Let stand 30-45 minutes, tossing occasionally. Add fruit with juices to a medium saucepan and cook over medium heat, stirring frequently, until softened, about 15 minutes. Let fruit cool slightly, then add to a food processor and puree until nearly smooth, leaving a bit of texture. Set aside 1/3 cup fruit mixture to swirl into ice cream; return remaining fruit mixture to saucepan.
Whisk together cornstarch and 3 tablespoons half-and-half in a small bowl; set aside. Add remaining half-and-half, sugar, Amaretto, and salt to saucepan with fruit mixture; bring to a boil over medium-high heat while whisking constantly. Whisk in cornstarch mixture. Return to a boil and cook for 1 to 2 more minutes, stirring until thickened. Remove from heat and cool to room temperature, then cover and chill for 6 hours in the refrigerator.
Pour chilled ice cream mixture into frozen cylinder of ice cream maker; freeze according to manufacturer’s instructions. Spoon half of the ice cream mixture into a freezer-safe container, top with dollops of fruit mixture, and repeat. Swirl layers together with a wooden skewer. Freeze mixture overnight until firm.
