EDIT: The question was about countably infinite set, not an infinite set, so this post does not answer the original question. I've decided to leave this answer here anyway, since it might still be interesting for the OP if he wants to know more relation of various versions of Ramsey Theorem to AC.
You can probably find some results about strength of Ramsey Theorem as a Choice Principle in Halbeisen's book Combinatorial Set Theory, Chapter 5, where several related Choice principles are mentioned.
A pdf-file with draft version of this book can be found at the website of the course on set theory he is teaching. (I'd say that version is very close to the final version, which was published.)
I'll quote some relevant results, proofs and more details can be found in the book.
$C(\aleph_0,\infty)$: Every countable family of non-empty sets has a choice function (this choice principle is usually called Countable Axiom of Choice).
RPP: If $X$ is an infinite set and $[X]^2$ is 2-coloured, then there is an infinite subset $Y$ of $X$ such that $[Y]^2$ is monochromatic.
Theorem 5.17. $C(\aleph_0,\infty)$ $\Rightarrow$ RPP $\Rightarrow$ KL $\Rightarrow$ $C(\aleph_0,n)$.
EDIT 2: After I learned that I originally wrote about a different version of the Ramsey theorem, I tried to check whether the same book mentions somewhere this version and - as expected, it does. Again, I've copied here some relevant parts, starting from p.11.
Ramsey proved his theorem in order to investigate a problem in formal logic, namely the problem of finding a regular procedure to determine the truth or falsity of a given logical formula in the language of First-Order Logic, which is also the language of Set Theory (cf. Chapter 3). However, Ramsey’s Theorem is a purely combinatorial statement and was the nucleus—but not the earliest result—of a whole combinatorial theory, the so-called Ramsey Theory. We would also like to mention that Ramsey’s original theorem, which will be discussed later, is somewhat stronger than the theorem stated below but is, like König’s Lemma, not provable without assuming some form of the Axiom of Choice (see Proposition 7.8).
Theorem 2.1 (Ramsey's theorem). For any number $n\in\omega$, for any positive number $r\in\omega$, for any $S\in[\omega]^\omega$, and for any colouring $\pi\colon{[S]^n}\to r$, there is always an $H \in [S]^\omega$ such that $H$ is homogeneous for $\pi$, i.e., the set $[H]^n$ is monochromatic.
The proof is done by first proving the case $n=2$:
Proposition 2.2. For any positive number $r\in\omega$, for any $S \in [\omega]^\omega$, and for any colouring $\pi \colon [S]^2 \to r$, there is always an $H \in [S]^\omega$ such that $[H]^2$ is monochromatic.
The proof uses Infinite Pigeon-Hole Principle, but it is only needed for countable infinite sets in this proof.
Infinite Pigeon-Hole Principle. If infinitely many objects are coloured with finitely many colours, then infinitely many objects have the same colour.