Judge Gregory L. Heath, who was appointed to his position by the former governor in 2009, dismissed Johnston's petition last week.
The 23-year-old filed for equality custody of Tripp, saying the 4-year-old should spend his time equally with mother and father, Radar Online reported Thursday.
Bristol Palin said Johnston was $67,000 behind in child support and wasn't a fit parent because he didn't visit Tripp on a regular basis. Sarah Palin -- the Republican nominee for U.S. vice president in 2008 -- has accused Johnston of being a "deadbeat dad."
Celebrity Breakups and divorces of 2014 [PHOTOS]