The homebound Mr. Grammarfusss...
 

...has to admit that he has doubt as to whether he should use "on the continent" or "on the Continent" to describe events in France, Germany and the like while excluding Britain and Ireland.

Your advice is welcome.

Definitely "on the Continent". Bon voyage!

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.

It doesn't help, because of the shortcuts of journalese, but it brings to mind the spoof headline from donkeys' years ago:

"Heavy fog over English Channel. Continent isolated."

Just out of curiosity, is Mr. Grammarfuss now homeward bound or housebound?

Mr. Grammarfuss is actually housebound now after suffering a severe case of vertigo on Monday morning while--highly inconveniently-- going on the air for his monthly gig on WGBH radio. After the benefits of an overnight stay at Mt. Auburn Hospital, he feels like has been through a war, but is feeling much better now and more grateful than ever for gifts taken for granted, like balance.

Oh, dear. Go carefully.

And bear in mind the sign at the ferry port, reading "Harwich for the Continent", to which some wag had added "but Paris for the rest of us".

My trainer makes me do coordinated head and eye rolls (leading with eyes) to strengthen vestibular system. Walking on cobblestone and lumpy brick streets, which with my neighborhood abounds, is good for balance practice too.