No, eCart will not manage stock for you automatically. Stock considerations need to be added manually.
The way that I have typically added stock management to eCart sites is to have a stock field and a stock update date field in the database.
Then since eCart will log any sales you can use SQL to subtract sales after the stock update date from the available stock without having to adjust the checkout process or what is stored during it.
Then you just need to update your product results, details, and shopping cart pages to account for out of stock items however you want to. Either not adding it, not showing it, or adding it with a message that delivery might be delayed.