>Is America falling behind the arms race?
Yes but not entirely because of Europe. Instead because of Israel and partially Europe too.

The world is becoming more Asian. Asia is becoming much richer and there's nothing anyone can do about it. South America and Europe are declining parts of the world economically. Africa has a booming population but is still dirt poor and will be for some time.
Asia is the future. Now - this also means future conflicts will be Asian. Not European. Not Middle Eastern either. Yes I know the Middle East is Asia but stick with the story. We will see some kind of resolution to the Korean dilemma this century. And the Taiwan dispute. As well as islands in the South China Sea. Kashmir? Why not. Maybe India will dam some of those rivers - think GERD on steroids. The Philippino claim to Sabah. Anyone with a dispute with Indonesia could stir shit up in Papua with proxies just to fuck with Jakarta.
This isn't an exhaustive list and I didn't even include the latest clash between Thailand and Cambodia. All of these conflicts can be resolved militarily and will require weapons. Who is building armies to fight in Asia? Not Europe. Not Israel. Not the US. And the former two keep trying to get the US back to the Middle East to sort out Gadaffi, Assad, Hamas, Hezbollah, Iran - all these conflicts which aren't part of the world global majority who already make a substantial part of the world's wealth and will soon own most of it in some decades' time.