Drain peaches, reserving syrup. Pour reserved syrup into a small saucepan and bring to a boil over high heat. Boil syrup rapidly, stirring often, until liquid has reduced by half, about 5 minutes. Remove from heat, pour into a bowl and cool.
Place raspberries in a food processor or blender; pour in reduced syrup and lemon juice. Process until sauce is smooth. Strain sauce through a fine-mesh sieve, pushing hard on solids to extract as much liquid as possible. (You should have about 2 cups of sauce.)
Arrange 1 or 2 peach halves on individual plates or in small serving bowls. Top each with a large scoop of ice cream. Spoon some raspberry sauce over peaches and ice cream. Pass around remaining raspberry sauce on the side.