Does entropy apply to Newton's First Law or does "acted upon" always require an external factor? 
First law: Every body remains in a state of rest or uniform motion (constant velocity) unless it is acted upon by an external unbalanced force. This means that in the absence of a non-zero net force, the center of mass of a body either remains at rest, or moves at a constant speed in a straight line.



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*Wikipedia — Newton's laws of motion
Doesn't the law of increasing entropy affect all objects though, since they are all in the closed system of the universe at large, and therefore they are all subject to slowing down, regardless of the containing medium, given enough time?
I guess what I'm curious is, can there ever be a body that will remain at uniform motion or uniform rest given that entropy must increase?
 A: Yes, it will (in the classical picture) continue forever, even with its entropy increasing, the entropy increase just means that some of the potential energy within the body will turn into heat (=kinetic energy), however the center of mass motion is unaffected.
Entropy increase says nothing about slowing down, rather the opposite.
A: This question seems to me harder and more interesting than either of the answers take into account.  A) GR has an analogue of Newton's First Law: the geodesics. B) The OP doesn't wish to consider the Universe as a whole.  @Jerry is taking the OP in reverse.  The OP is: since none of the usual bodies we study is the whole Universe, none of them are closed systems, therefore friction and dissipation effects exist, and so, does this imply that the Second Law of Thermodynamics trumps Newton's First Law...considering either of them as approximations governing the evolution of the body in question over a very long time-scale.
Because the time-scale is very long, actually the phrase used in the OP is «given enough time», it is not clear to me that we can depress the interactions of the body in question with the rest of the Universe sufficiently.  The numbers must be crunched at this point....
Okay, here is the answer, although not the complete answer: For a fixed body that we wish to study, no matter how well-insulated and isolated it is (in reality....therefore it cannot be perfectly insulated and isolated), eventually the effects of dissipation with the environment will make Newton's First Law a poor approximation for the behaviour of that body because it's hypotheses will not be fulfilled.  The crucial hypothesis is the absence of an external force.  But dissipation is in fact due to fluctuating external forces.  AS the Universe as a whole approaches statistical equilibrium, every body in it will behave like a Brownian particle with a random, non-differentiable motion.  In fact, even worse, the «body» will cease to be a separate body before this point is reached...so the unspoken but implicit hypotheses of Newton's Laws will also break down.
At the moment, the Universe is far from statistical equilibrium, and it is unlikely to ever reach it if theories of the «big crunch» are correct, but what I sketched above might hold good for local parts, given enough time.  So the complete answer would depend on: how big is this body?  Will it disintegrate before the end of the Universe or not? Is there a local region in which it will remain for long enough for statistical equilibrium to be reached?  Or if not that, something else?  
I.e., the question is about the old idea of «heat death» of the Universe, but in light of present-day cosmology, the old Boltzmannia--Nietzschean conclusions are not well-established.  And the question is about partial «heat degenerations» of a local part of the Universe, degenerations not really deaths but bad enough to make Newton's Laws imperfect tools for studying the body in question.  Cf. Feynman's discussion of «ratchet and pawl» in his Physics Lectures, and also recent work on Szilard engines.
A: If you ignore the microscopic explanation of entropy, entropy is just an internal state of a system, on par with the system's volume, or the number of particles in the system.  If you have a gas with a fixed entropy (let's store it in a vacuum tube so it doesn't escape, and let's give the tube infinite insulation so it doesn't leech out any heat) out in deep space, and you throw it, it will just happily trail off in a straight line forever with a constant entropy, volume, and number of particles.  
You get changes in entropy and whatnot only when the system in question interacts either with another system or its environment.  But it really is best thinking of these things, at least on a macroscopic scale, as internal degrees of freedom of the system.
