Please go to their web site and use their automated response (sort of an interactive FAQ).

Type in trip cancellation or what you think is pertinent.

Follow the links from there to see what a cancellation vs. a postponement to another flight within a year would cost.

It all depends on the terms of your ticket. I doubt there would be a refund. You'll likely have a cancellation fee that will be deducted from the cost of the tickets. You can then use the remaining amount to apply toward other tickets within a year. United can give you the definitive answer, but that's usually how it works.

I hope things are okay with your wife.

Did you buy trip insurance?

The best thing to do is to call United and explain your situation. Escalate it to a supervisor if you hit a wall early on. Most often than not they will just allow you to use the credit of the flight up to one year. This is actually what happens if you do a no show. Say you paid 1200$ then you an use your $1200 credit for up to one year from the date of purchase but they will charge you a $150 change fee and also a difference in fare. There is no cancellation fee or no show fee but make sure you call them and cancel it. Sometimes they may wave this change fee.
The restriction on your credit however will be that you have to fly it in the same market. USA to Southern South America or Northern South America. You can not use it to fly on a domestic USA or even a Europe trip.
If you plead and you cry they may make a compassionate exception but this is rare. Don't give up, be nice and polite and that may help you a long way.

I don't think it will do any good to plead and beg unless you have doctor's orders not to fly you can send them. They hear excuses every day from people wanting to changes flights and avoid the change fee.