No. Receiving benefits on the earnings record of your ex-spouse will not change what that person can receive from Social Security. They'll collect the benefit they're entitled to, regardless of whether you claim an ex-spousal benefit.
The same holds true for any other benefits paid on your former mate's record. If, for example, your ex has remarried, the new husband or wife could draw spousal benefits, and they will not be affected by your divorced-spouse benefit.
Keep in mind
If you remarry, you will almost certainly lose any benefits you are collecting on your former spouse's record. For the purpose of divorced-spouse benefits, your ex's marital status does not matter, but yours does.