Yes, spellcasting is automatic as long as nothing impedes (bound or gagged caster) or interrupts (silencing, being struck or distracted, etc.) the casting attempt. In the "Spell Casting" section it states this:

"The sorcery of magicians requires both verbal incantations and complicated hand gestures. Sorcerers who are prevented from speaking or from moving their hands are unable to cast spells. Weapons must be sheathed, shields shouldered, and other handheld items dropped or stowed. The sorcery of clerics is less restrictive; because their sorcery is almost exclusively verbal, they are able to maintain their weapons and/or shields whilst casting. If a spell caster is struck before his spell fires, his spell is disrupted."

Thanks, all.  I was a tad confused because it didn't seem to spell out implicitly that the spell goes off automatically unless the caster is somehow interfered with or the spell has some kind of qualifying roll associated with it.  I wanted to ensure I was understanding it correctly.  I appreciate it.

Well, I did used to play red box D&D and AD&D but it was many moons ago.  I also own Labyrinth Lord, Castles & Crusades, the Dungeon Crawl Classics RPG, etc.