Complement to the initial answer (below)
(added on September 4 2013)
My initial answer to this question (below) was only based on my
knowledge of syntactic and semantics issues in computing, as well as
some knowledge of various computing techniques such as computing on reals
with infinite precision, or more precisely with arbitrary precision.
Looking for a better understanding of the issue, I found that it is
currently actively researched. Though I have not explored much, I
found that my perception of the crucial role of denumerability, which
can be traced to the fact that everything is expressed syntactically,
hence with denumerable systems of symbols, or symbols conbinations,
is indeed justified.
One approach to prove Church-Turing thesis as a law of physics relies
on assuming a specific property of the physical world, presented as
dual of the limitation on the speed of light and information, which is
a limitation on the space density of information, both limitations
together ensuring density limitation in space-time. The translation of
this new law in physical terms can actually be subtle to account for
various existing physical laws. This apparently excludes unregulated
use of real numbers.
Giving a bound to the amount of information to be found in a given
volume of space-time seems to be the direct counterpart of the
fineteness of what can be done by a computing process in finite time,
and of the consequent denumerability of whatever may be considered
without setting a time limitation.
I am also addind a note at the end to explain why, depite their infinite tape, Turing machines must be considered a physical model of computation, and have been thus considered by most people, afaik.
Note: I try to give the best answer I can, but I am stretching my
confidence in my own understanding. The distinction between
mathematics and physics is a topic I find fascinating (though I do not
have much to say about it), and it is at
the heart of the question, imho.
I somewhat disagree with Ben Crowell's interpretation of Church-Turing
thesis. If it were the observation that a given variety of models of
computation are all equivalent, it should be a mathematically provable
hypothesis, or at least we could hope for a proof. This is not the
case because the thesis states indeed the a function is computable
according to whatever model of computation only if it is computable by a Turing
machine (hence by any Turing complete model).
However this thesis is indeed motivated by the fact that all of the
many models of computations that were designed by mathematicians and logicians
turned out to be equivalent (a few being a bit weaker).
It is true that the Turing machine has an infinite memory tape, which is not too physical. This
is not really essential since the interesting results are those
computable in finite time, thus using only a finite part of the
infinite tape. The tape is chosen to be infinite because it is not
known in advance how much will be needed.
The infinity of the tape becomes a problem in itself
only when envisionning the possibility of infinite time, for example
in Closed timelike curves.
Many "non physical", or "not yet physical" (?) models of computation have been
considered by mathematicians, which could become "physical" (that is,
"operational") if ... (see hypercomputation).
An interesting example is also oracle machines:
what can we compute if
we know how to solve such and such problem (not Turing computable, of
course). If some physical breakthrough actually allowed to solve this
problem, the problems addressable by oracle machine would become computable.
The wording of the previous paragraph actually means that, despite its
infinite tape, the Turing machine is actually viewed as a physical
device (by computer scientists if not by physicists), even if somewhat
idealized. This is probably
because a computer scientist considers that an infinite object is
computable, or definable, if it is the limit of a sequence of finite
objects that are all computable. A machine with infinite tape may be
seen as the limit of a sequence of finite tape machines.
In this sense, we can say that Turing computability is a consequence
of the laws of physics, since these law allow us to make computers,
and we can in principle add memory as needed, for the time we are
willing to wait for the result. This is no worse an approximation than
physics theories have proved to be so far, given enough time to
find out that they only approximate physical reality. This is also no
worse than using calculus to reason about phenomena that are
ultimately discrete rather than continuous.
But this is not the question being asked. The actual question is
whether the Church-Turing hypothesis can be derived from the laws of
physics. This thesis is that there is not a more powerful model of
computation. If all phenomena describable by existing physical
theories can be simulated by a Turing machine (a computer program),
that does indicate that there is nothing in these theories that allows
for more powerful computing models. But can they be simulated ?
An important chracteristic of Turing-Church computability is its discrete
character (this is actually true of all human intellectual activities). It is mostly what people call derisively "symbol pushing".
What can be computed is essentially discrete and denumerable. Physics
theories are usually express as continuous structures. Are they, or is
it just a convenient representation ? If they are continuous, it is not
obvious that they can be simulated by a computer, even though
computers can to some extent deal with continuous entities, such as
real number, finitely represented, but there are all kinds of
limitations to these computations and their uses.
Simulating discrete approximations of these theories is probably not
satisfactory, and seems (to me at least) a bit tautological, saying
that we can simulate with a Turing machine that part of physics that
is simulatable by a Turing Machine (because it is discretized).
Hence I do not see as very convincing the existence of computer
programs that simulate known physics theories, and I do not much believe in the reducibility of physics to computing.
But is that necessary to make the Church-Turing thesis a law of physics ?
Furthermore, the Church-Turing thesis is only an hypothesis. We do not
know whether it is actually true. But it is true as far as we know,
like laws of physics.
To push the problem further, there is a result called the Curry-Howard
isomorphism, that shows that computer programs (read "Turing Machines")
and mathematical proofs are fundamentally the same, in the sense that
they have identical axiomatisation. Hence, for any statement you wish to
make about Church-Turing computability, there is an equivalent
statement about mathematical proofs.
So the question might also wonder whether our very exclusive way to do mathematics
since Euclid's Elements can be viewed as a consequence of a law of physics. And it
could restate the last point as: can it be deduced from fundamental
laws of physics that mathematical theories must be constructed as we
do it. And that applies naturally also to physical theories. Now we
get a bit in a loop that I will not try to sort out.
Note about the physicality of Turing machines despite their infinite tape.
(added on September 4 2013)
The fact that Turing Machines (TM) have infinite tape cannot be
construed as evidence that they are not a physical model of
computation. The infinite tape is only a mathematical device to
simplify the analysis of computability, but whatever they compute only
uses a finite tape.
The whole theory could be built on Finite Turing Machines (FTM), i.e.,
Turing Machines with finite tape, which are actually just finite state
automata, the simplest kind of of formal computing device there is.
These are definitely easy to implement as physical devices. FTM have a
special state they enter when they run out of tape, called MemOverflow.
Then we consider classes of FTM, such that all FTM that differ only by
the length of their finite tape belong to the same class.
Let C be such a class. We say that C halts on an input x if there
is a FTM M in the class C such that its tape is long enough to be
initialized with x, and the computation of M with its tape thus
initialized never loops (which is easy to detect on a finite state
device) and never enter the state MemOverflow. If there is no such
machine M in C, then C is said to not to halt on this input.
For all intents and purposes, there is just no difference between these
classes of finite state machines and Turing machines with infinite tapes. Doing the theory
is just a bit more awkward for no benefit.