If you add just a small amount of dilute nitric acid, then you'll get the compound Cu(NH3)4(NO3)2, but please be VERY careful with that. This compound is a high explosive and just a gram of it can blow off your fingers.
If you add much more dilute nitric acid, then the tetrammine cuprate ion is broken down. First, you'll get some copper hydroxide again and on adding more acid, you'll see that all dissolves and you get a light blue solution of copper nitrate and also (colorless) ammonium nitrate.
If you want to play with the copper ammonia complex, then just don't use the nitrate ion as counterion. Use the sulfate ion instead (i.e. sulphuric acid). Use of the chloride ion (i.e. hydrochloric acid) also is not a good choice, because chloride also forms complexes with copper and that certainly will interfere with the ammonia complexes.