To make the cherry syrup, combine the water and sugar together in a 4-quart saucepan and bring to a boil over high heat, stirring occasionally.
Place the cherries in a sachet and close the top. If you do not have these ready-made sachets (much like coffee bags), you can use cheesecloth. Or omit the bundling step and simply strain the cherries later.
When the sugar has dissolved, add the cherries and lime juice, reduce the heat to medium-low, and simmer, uncovered, until you have a pale, cherry-infused syrup, about 20 minutes. Set aside to cool.
Remove the cherry sachet or strain the mixture, squeezing the juices from the sachet into the saucepan or pressing on the solids in the strainer with the back of a spoon. Store the syrup in a glass jar at room temperature or in the refrigerator until ready to use.
To make a cherry spritzer, dilute the syrup with water, seltzer, or sparkling wine, using a rough ratio of 3 parts water or wine to 1 part syrup. Serve over ice, if desired.