I don't see why not - it may cost you an arm and a leg though...

Could you purchase it by phone?

Yes, of course you can purchase the ticket at the airport. Most airlines have separate counters where they sell tickets (as opposed to checking in passengers). Since United and American both count Chicago O'Hare as major hubs, I don't see any problem with this other than cost. You'll probably pay in excess of $1000 for the flight and may undergo a lot of extra security screening if you purchase a one-way ticket in cash.

Just make sure you have enough time between flights. I'd say minimum of 3 hours, perhaps 4. If you are arriving in the morning, there are probably plenty of flights in late morning and early afternoon you could catch, but there's always the chance that they will be booked on your day of arrival. With all airlines flying at max occupancy on some days, this can happen.

Frontier flies to Denver, it's home airport, from Chicago Midway, so that probably won't be of much help to you, but other airlines besides United and American may fly this route.

Yes, you can - but

1) the price will probably be outrageous
2) you will definitely undergo much more stringent scrutiny from security - buying a one-way ticket is a no-no and so is buying one the day before. Get to the airport extra early to allow for a long time with the security people.

Thank you very much for all the information !

All the advise above is significant. Air fares increase at 24 hours, 72 hours, 7 days, 14 days and 28 days. You could purchase from a local travel agent or if you are in Paris as in your name United has a company owned ticket office in Paris with cash or as you prefer and minimize your expense and security scrutiny.

Purchasing in cash at the airport counter same day or within 24 hours could result in a lengthy security interview or more attention to you and your belongings than you would like.