Oh it's far more complicated than that.
The player is eligible to come off the injury DL sometime between July 6th and July 15th.
When a player is placed on the DL it's retroactive to the last day he played. So if the player was placed on the DL 10 days after he had last played (the limit for retroactive DL placement), he might be eligible to come off in just 5 days. Generally this is suboptimal, but teams will often let a player sit a few days to see if he can still play before placing him on the DL.
That said, the number of days is counted from the last date that the player played. Players are routinely listed as placed on the DL the day after they are hurt. So if you're hurt the last day of June and placed on the DL on the first day of July, you are eligible to come off the DL on the 16th of July.
This is illustrated in a data dump from baseball heat maps. Looking at 2013 specifically, we can see only a few players who were only on the DL for 15 days....