Pour the water into a saucepan, and bring to a boil; stir in sugar until dissolved. Bring the syrup back to a boil, reduce heat, and boil gently until the syrup is reduced to 1 cup, about 10 minutes. Remove from heat, and allow to cool.
In a mixing bowl, mash the avocados to your desired consistency, either smooth or chunky. Mix in the soy creamer and lime juice. Stir in the cooled sugar syrup until completely incorporated. Chill the mixture completely in the refrigerator, about 4 hours.
Pour the chilled mix into an ice cream maker, and freeze according to manufacturer's instructions; pack the ice cream into a lidded container, and freeze until firm, at least 1 hour.