tl;dr: not really.
Viable to whom?
To you and me? Sure: we know who won in the end, and we risk nothing, having been born long after the war.
To Churchill? Sure: he was committed to fighting to the end, and he would have welcomed any help.
But for a french soldier? No. The year is 1940. Germany seems unbeatable: it was starved into a defeat in WW1 by the naval blockade, but this time Russia is clearly on the German side - a friendly non-combatant if not an outright ally, supplying Germany with food and materiel. Your family is either in the occupied part of France or in the Vichy zone; either way you put them in harms way and you will be branded a traitor, not a hero. When the Germany inevitably wins and you fight for the wrong side, you will never be able to go home.
Besides, fight for what? For Poland? For the Jews? (Hitler always defined his war as "the war on jews").
So, very few people followed de Gaulle.
This attitude was sealed by the Attack on Mers-el-Kébir that
was almost universally condemned in France and resentment festered for years over what was considered a betrayal by their former ally
What was viable?
French Navy was huge and vitally important to both sides.
Churchill even offered to release France from the obligation not to make a separate peace (signed in March 1940) in exchange for the ships, or to create a Franco-British Union State.
So, if the French gave their ships to Britain (with or without sailors), that would have been an important development for the anti-nazi alliance (which at that moment consisted of Britain and a bunch of governments in exile in London).
Of course, the French colonial industrial resources were virtually non-existent, but that was not the issue: the French would have been supplied by Britain and the US. In fact, the aforementioned Attack on Mers-el-Kébir was intended, in part, to demonstrate to the US that Britain was committed to the war with Germany and that the American aid will never fall into German hands. Thus, if the French navy joined the British (the French overseas ground forces were relatively insignificant - and it is unlikely that many troops agreed to evacuating from Europe to Africa to keep fighting, see above), both "Arsenal of Democracy" and Lend-Lease would have probably came earlier.
The French had the capacity to keep fighting, at least on the high seas, what they lacked was will.