Assuming the question is "will they close it for inactivity (alone)".. the answer is "Nope"
Update: < My answer is geared to credit Cards issues by companies that deal in credit, not merchandise (i.e. store cards, retailer cards, etc). Retailers (like Amazon, etc), want to sell goods and are in the credit card business to generate sales. Banks and credit companies (about whom I am referring) make their money primarily on interest and secondarily on service charges (either point of use charged to the vendor that accepts payment, or fees charged to the user).>
The only major issuer I will say that it might be possible is Discover, because I never kept a Discover card. I also don't keep department store cards, which might possibly do this; but I do doubt it in either of those cases too. My answer is based on Having 2 AMEX cards (Optima and Blue) and multiple other Visa/MC's that I NEVER use... and most of these I have not for over 10+ years.
Since I am also presuming that you are also not talking about an account that charges a yearly or other maintenance fee..
Why would they keep the account open with the overhead (statements and other mailings,etc)?
Because you MIGHT use it. You MIGHT not be able to pay it off each month. Because you MIGHT end up paying thousands in interest over many years. The pennies they pay for maintaining your account and sending you new cards with chip technology, etc.. are all worth the gamble of getting recouped from you! This is why sales people waste their time with lots of people who will not buy their product, even though it costs them time and money to prospect.. because they MIGHT buy.
Naturally, there are a multitude of reasons for canceling a card; but inactivity is not one. I have no less than 10+ "inactive" cards, one that has a balance, and two I use "infrequently". I really would not mind if they closed all those accounts.. but they won't ;)
So enjoy your AMEX knowing that your Visa will be there when you need/want it.. The bank that issues your Visa is banking on it! (presuming you don't foul up financially)