I married a man I met while traveling in Europe, and sponsored him for a U.S. green card. Stupid me, I must have been blinded by love, because I hardly read what I was signing. Now we're divorced, and I discover that by signing Form I-864, I was promising to keep supporting him even after the marriage is over. So, he's already told me he plans to make me send him money while he continues living here in the United States. I'm furious – he's perfectly capable of going out and getting a job. He even speaks English. Shouldn't he have to try before I'm forced to step in and send him money?
If it's any comfort, you're not the first to find yourself in such a situation. There are only a few ways in which a U.S. petitioner who signs an I-864 Affidavit of Support can get out of the obligation to support the immigrant at an amount that's 125% or more of the U.S. Poverty Guidelines levels each year. These include if the immigrant becomes a U.S. citizen, works for 40 quarters (in Social Security terms – that's approximately ten years), dies, or permanently leaves the United States.
Assuming none of these happens anytime soon, your ex would likely have to sue you to make you pay up on this obligation. At that point, other U.S. petitioners have tried arguing to the court that (as you suggested) the immigrant should have to show that he or she is seeking employment before turning to the sponsor. However, this argument has so far failed (as of late 2013).
What usually happens next is that the U.S. sponsor settles with the immigrant, by offering a flat payment of cash. This has the advantage that you avoid making ongoing payments and possibly returning to court at later dates to determine whether your obligation to your ex is still in force.
As you can see, speaking with a lawyer would be an excellent idea in this case, both to double check on the latest court cases regarding the immigrant's obligation to "mitigate damages" and look for a job, and to help negotiate a settlement or other solution.