If modern computers aren't actually Turing-complete, does that mean that it is possible to determine if a program run on such a computer halts?

The halting problem says that it is impossible to create a general algorithm which can for all inputs and programs determine whether they halt. However, this assumes that the programs and/or the things running them are Turing-complete.

Since all computers have a finite amount of memory and are therefor not Turing-complete, does that mean that the halting problem does not apply to a program run on a machine with finite memory?

You can take a look at:
https://en.wikipedia.org/wiki/Halting_problem#Common_pitfalls

One can realize that a program is not halting by seing that it's repeating its behavior.

"A machine with finite memory has a finite number of states, and thus any deterministic program on it must eventually either halt or repeat a previous state"

It also says that even computers are finite state machine, it's more useful to model them as infinite state machines, just because the number of states it's huge.

"Minsky warns us, however, that machines such as computers with e.g., a million small parts, each with two states, will have at least $2^{1,000,000}$ possible states"

StackExchange Computer Science Q#63055, answer score: 6