The title and the question ask something different.  "Is X sufficient" versus "is there a single instance of Y".  Some examples of Y have been proferred.  

A very mathematical approach, find a single counter example.  However the question is sufficient is actually a value question.  One does not have to (I would argue should not) accept the idea that one needs to know everything.

Looking at the title, a little more holistically, I would actually answer "pretty much so" to the question of if R integration is sufficient.

Let's think about it practically:  are there any homework problems in the following courses, that you need L integration to solve?

Undergrad:
-calc-based intro physics?  No.
-junior year E&M? No.
-junior year mechanics? No.
-"modern physics"? No.
-quantum theory?  No.
-thermal physics? No.
-undergrad electives in optics, quantum optics, acoustics, nuclear, sonar, solid state, etc.  No!

Grad school:
-Jackson E&M? No.
-Mechanics? No.
-Quantum?  No.
-99.44% of all the other courses?  No.

So...you can learn a heck of a lot of physics, get a degree, and have a lot of commercial value without learning L integration.  Note, how this is different from other concepts that you do need at various stages (integration by parts, PDEs, etc.)  Compare how dramatically different the use of R integration is with the abstract examples offered up here.

P.s.  I have done a lot of USE of probability and statistics in physics, chemistry, engineering, business, and psychology.  And I did not need L integration (I don't know it!)