No. Google has the monopoly on search (outside of China). Avoiding anti-trust is a huge concern of theirs. Look at MS and IE in the browser wars - they were forced by the EU to build into Windows links/ads to the other browsers - see https://en.wikipedia.org/wiki/BrowserChoice.eu#/media/File:B...
Whilst DDG has huge momentum by now, duckduckgo.com is still an ache to type - that's why ddg.co has existed for years, and why you use "DDG" yourself ;) duck.com is way more memorable and even faster.
Not a lawyer, but given DDG's brand recognition, I'd imagine they would have a strong case in an ICANN dispute given that Google are not using it. Google are proactively avoiding at least this bad PR by doing the right thing (this week).