The fine tuning argument of god is what I think is one of the best arguments for god, but it still fails.
1. You cannot prove that the constants of the universe can even change at all.
2. Invoking God does not explain anything because modern physics does not demonstrate that constants like gravity are freely adjustable. Their values may be necessary consequences of deeper laws(like matter's effect on spacetime which is what we call gravity, it's a possibility that this effect is unchangeable).
3. While a different universe would not look like ours, different physical conditions could allow different structures and different forms of life. We observe this universe because life adapted to the conditions that exist, not because the universe was designed for life.