To use the Ledger Nano S as a standalone wallet then yes, you need either Google Chrome or a browser which is compatible with Chrome, such as Chromium.
However, it’s also possible to use the Nano S with other wallets, including webwallets (such as MyEtherwallet), mobile wallets (eg. Mycelium) and software wallets (eg. Electrum).
You will still need Chrome or compatible browsers to initialise the Nano S (and perhaps to upgrade its firmware too).