Microsoft Word insists that it should be but I don't know why.
The word ethernet is capitalized because Ethernet is the currently accepted convention.
Longer—but still not entirely satisfying—answer:
The reason Ethernet is the current convention remains unclear. Authoritative explanations are absent or inconsistent. It cannot be explained by trademark status or by comparison to similar examples. Nevertheless, the convention itself is surprisingly consistent.
Ethernet does not appear to be an active registered trademark. (Disclaimer: I'm not well-versed in trademark application procedure or terms, but the following is what I could gather.) Application for ownership of the mark ETHERNET was filed on behalf of Xerox Corporation in 1979 and registered in 1981. The trademark applies to goods classified as "apparatus and instruments," described in the application as "Electronic Communications Networks, Transmitters, Receivers, Cables and Controllers." The trademark was cancelled in 1988 after Xerox did not file a declaration under Section 8 of U.S. trademark law, which requires an affirmation stating the mark "is in use in commerce." Subsequent to Xerox's lapse in ownership, two applications to trademark ETHERNET have been filed. Unlike Xerox's trademark, which applied to goods, the newer filings applied to services. The applications were made on behalf of different LLCs with the same physical address, the first in 2011 for "Provision of access to the internet" and the second in 2013 for "Telecommunications services, namely, Internet access provider services."* Neither trademark is in force: both applications have a status of "Dismissed or Invalidated" because of failed or late responses to the Trademark Office.
In any case, the existence of a trademark for Ethernet would not necessarily explain its capitalization. Trademarks can become genericized, even in the face of corporate strategies to counteract genericization. Wikipedia's list of generic and genericized trademarks, for example, lists Xerox® and Kleenex® under "protected trademarks frequently used as generic terms." A quick internet search of style guides suggests Xerox and Kleenex are commonly used as examples of trademarks that should be capitalized. The section on trademarks (§8.153) in The Chicago Manual of Style lists both as examples of trademarks that should be capitalized (if their use cannot be avoided), but also notes that Webster's differentiates the verbs xerox and google, among others, which are not capitalized. However, in Microsoft® Word with AutoCorrect™ enabled, typing
use kleenex on the xerox copier is corrected to
Use Kleenex on the xerox copier. Word does not appear to capitalize any use of xerox, independent of part of speech, including
xerox corporation. Thus, even for these two widely cited examples, agreement is not universal.
Compare to I/internet
The words internet and web have undergone a change somewhat similar to ethernet, though the conventions are not settled and there isn't consensus on what explains the changes. One style guide for online writing specifies Ethernet, without explanation, while noting that "Per [Associated Press] Stylebook 2016, internet is no longer capitalized." (This same guide capitalizes Google, Googling, and Googled.) The SUSE style guide insists on Ethernet; the guide gives reasons for rejecting some terminological variations (e.g. "Ethernet card" is preferred to "wired card" because the latter "sounds as if wires attached to the card are meant"), but does not specify the reason for preferring Ethernet. Chicago's section "Terms like 'web' and 'internet'" (§7.80) maintains that "Ethernet (a trademark)" should be capitalized. Unlike the cases of Xerox and Kleenex, agreement on Ethernet seems near universal, though explanation is inconsistent. And while Chicago explains Ethernet with reference to its trademark status, it could just as plausibly now be written ethernet given that it has been effectively genericized and, if my research above is correct, it's no longer a registered trademark.
Does Ethernet "need to be" capitalized? Apparently not. Xerox did not renew its trademark on ETHERNET though presumably it could have, since the instruments in the trademark application were in commercial use at the time of its cancellation. Had Xerox renewed the ethernet trademark, it could have campaigned against genericization, as it has done with Xerox. The case of ethernet appears to be an unusual one in which the mark's capitalized form, denoting intellectual property, has been retained despite its de facto genericization and de jure loss of proprietary status.
Is Ethernet capitalized? Apparently so. Style guides tend to prefer it. Thus, according to Google Ngrams, Ethernet is by far the most frequently indexed variation, although others less common than ethernet (e.g., EtherNet, EtherNET) have also been indexed. Thus, style guides tend to prefer it, and so on. In other words, ethernet is capitalized because most writing capitalizes Ethernet. Why does Word correct ethernet? Because it's (usually) capitalized.
* Note the differing capitalizations of "internet" between applications, with the later one capitalized.