If my wife was given the wrong medication while in a nursing home, do I have grounds for a lawsuit?

Medical malpractice is negligence.  With regard to the nursing home, negligence is the failure to exercise due care (that degree of care that a reasonable nursing home would have exercised under the same or similar circumstances to prevent foreseeable harm).

If your wife suffered adverse consequences from being given the wrong medication, she has a claim against the nursing home for negligence.  If adverse consequences, injury, etc. did not occur, she wouldn't have much of a case against the nursing home.

Prior to filing a  lawsuit against the nursing home for negligence, it may be possible to settle the case with the nursing home's insurance carrier.  Your wife's claim filed with that insurance carrier should include her medical bills and medical reports.  Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of her injury/illness from being given the wrong medication, and will be used to determine compensation for pain and suffering, which is an amount in addition to the medical bills.

If the case is settled with the insurance company for the nursing home, NO lawsuit is filed.

If you are dissatisfied with settlement offers from the nursing home's insurance carrier, reject the settlement offers and file a lawsuit for negligence against the nursing home.  If the case is NOT settled with the nursing home's insurance carrier, your wife's lawsuit for negligence must be filed prior to the expiration of the applicable statute of limitations or she will lose her rights forever in the matter.

There are two different elements to a successful lawsuit: liability, and damages.

Liability is the obligation to pay compensation, and is based on fault: did the other party do something negligent, or unreasonable careless, violating their duty of care? Giving a nursing home patient the wrong medication may very well be negligent, and so could give rise to liability.

But there also must be damages, or some injury or cost, caused by the negligent act. If your wife did not suffer some injury from the wrong medicine, or incure medical costs (e.g. had to be taken to the hospital, and paid something out of pocket for it), there is no "damage" in a legal sense, and so you would not get any compensation. Even if there was some mild or modest damage--she was sickened for a few days; you had to pay $200 in a hospital co-pay; etc.--it would not be worth suing, since the amount you could recover would be so much less than the cost of the lawsuit (for this kind of case, you'd have to hire a medical expert, and they do not work cheap). The compensation you can get is related to the extent or severity of your damages. Only if your wife was greatly injured or incurred significant medical costs might a lawsuit be worthwhile.