Again, textual context is king. And queen is where the published work/audience is expected to be.
But this is what my
Oxford Dictionary for Writers and Editors advises.
Continent/, -al cap. when referring to mainland Europe as distinct from Britain, lower case when referring to countries that are also continents, such as Australia, continental USA is the USA excluding Hawaii but including Alaska; abbr. cont.
The Chicago Manual of Style (8.50) advises thus:
the continental United States; the continent of Europe, but on the Continent (meaning in continental Europe as opposed to Great Britain); Continental cuisine, but continental breakfast
Note the difference in prepositions.
The CMS also stresses "The following examples illustrate not only the principls sketched in 8.2-4 but also variations (not always logical) based on context and usage."
At the risk of seeming a pedantic boor, I will mention that one or more up-to-date style guides should be included among every writer's desk references. Though counsel was recently given in these illustrious halls/boards roughly that only ninnies worried about correctness, and real writers should fight empty convention.
Cross-posted with Cathy.