Combine the honey and water in a saucepan set over medium heat. Cook until the honey dissolves, then reduce heat to low. Bruise the rosemary with the back of a heavy knife, then add to the pot.
Cook 15 minutes. Remove from heat and let cool completely; discard rosemary.
To prepare the cocktails, muddle the ginger in the bottom of a cocktail shaker. Fill the shaker with ice and add the two juices and 2 ounces of the rosemary-honey syrup (you'll have some left over). Shake well, until chilled. Stir in the gin. Strain into 4 ice-filled glasses. Top with club soda. Add one lime wedge to each drink; garnish with a rosemary sprig.