It's not the deficit, no. As you correctly stated, the definition of deficit doesn't involve GDP. You could measure deficit as a proportion of GDP to compare deficits across differently-sized economies, but that'd be a different statistic. For example, imagine:
Australia has: GDP $1,000; government revenue $500; government expediture $600
New Zealand has: GDP $500; government revenue $200; government expediture $300
Both have a deficit of $100 per year. If they have been running the $100 per year deficit for ten years, both have $1000 in debt. For Australia that's a 200% debt -to-revenue ratio, but for NZ it's 500%.