If you were designing a programming language that features automatic memory management, would using reference counting allow for determinism guarantees that are not possible with a garbage collector?
Would there be a different answer to this question for functional vs. imperative languages?
In real time programming garbage collection could be harmful, because you don't know when the garbage collector will collect... so yes, reference counting is definitely better in this context.
As a side note, usually in real time system only some parts needs real time processing, so you could avoid garbage collection just in sensitive components. A real world example is a C# program running on a Windows CE target.
If you look at the RTSJ spec (JSR-1), you'll see they did an end-run around the problem by providing for no-heap realtime threads. By having a separate category of thread that isn't allowed to touch any object that might require the thread to be stopped for garbage collection, JSR-1 side stepped the issue. There aren't many RTSJ implementations right now, but the area of realtime garbage collection is a hot topic in that community.
From some involvement in various projects migrating significant chunks of code from C++ (with various smart pointer classes, including reference counted) to garbage collected Java/C#, I observe that the biggest pain-points all seem to be related to classes with non-empty destructors (particularly when used for RAII). This is a pretty big flag that deterministic cleanup is expected.
The issue is surely much the same for any language with objects; I don't think hybrid OO-functional languages like Scala or Ocaml enjoy any particular advantages in this area. Situation might be different for more "pure" functional languages.
would using reference counting allow for determinism guarantees that are not possible with a garbage collector?
I don't see how. The process of lowering the reference count of an object is not time-bounded, as that object may be the single root for an arbitrary large object graph.
The only way to approach the problem of GC for real-time systems is by using either a concurrent collector or an incremental one - and no matter if the system uses reference counting or not; in my opinion your distinction between reference counting and "collection" is not precise anyway, e.g. systems which utilize reference counting might still occasionally perform some memory sweep (for example, to handle cycles).
You might be interested in IBM's Metronome, and I also know Microsoft has done some research in direction of good, real-time memory management.