The definition of an $O$-module of Rotman textbook, where $O$ is a sheaf of comutative rings over a space $X$ is: an $O$-module is a sheaf $F$ of abelian groups over $X$ such that
(i) $F(U)$ is an $O(U)$-module for every open $U \subset X$;
(ii) if $U \subset V$, then $F(U)$ is also an $O(V)$-module, and the restriction $\rho^{V}_{U}: F(V) \rightarrow F(U)$ is an $O(V)$-module homomorphism.
It seems to me that it is equivalent to ask that each stalk $F_{x}$ be an $O_{x}$-module. Is this right? If so, how can I prove it?
Thanks in advance.
No, this is not just an $O_x$-module structure on $F_x$ for each $x$. For a really simple example, let $X=\{a,b\}$ with topology $\{\emptyset,\{a\},X\}$. A sheaf on $X$ is just determined by the two values $F(X)$ and $F(\{a\})$ with a map $F(X)\to F(\{a\})$, and $F(X)$ is the same as the stalk $F_b$ and $F(\{a\})$ is the same as the stalk $F_a$.
So, a sheaf of commutative rings on $X$ is the same as giving two commutative rings $O_b$ and $O_a$ with a homomorphism $O_b\to O_a$. A sheaf of abelian groups is two abelian groups $F_b$ and $F_a$ with a homomorphism $F_b\to F_a$. To make $F$ an $O$-module then is more than just giving $F_b$ the structure of an $O_b$-module and $F_a$ the structure of an $O_a$-module: additionally, the map $F_b\to F_a$ must be an $O_b$-module homomorphism with respect to these structures (using the homomorphism $O_b\to O_a$). In other words, we can't just have any two random module structures on $F_b$ and $F_a$; they have to be compatible with the restriction maps of $O$ and $F$ in a certain way.
More generally, giving an $O$-module structure on $F$ is determined by an $O_x$-module structure on $F_x$, but these can't just be arbitrary independent module structures: the assumption that they all arise from module structures over each open set is a severe extra restriction that imposes a certain "compatibility" between the module structures at different points.
Here is another example which you may find illuminating. Let $k$ be an algebraically closed field and let $X=k$ with the cofinite topology. This has a natural sheaf of rings $O$ defined by $O(k\setminus\{a_1,\dots,a_n\})=k[x,(x-a_1)^{-1},\dots,(x-a_n)^{-1}]$ with the obvious restriction maps. The stalk $O_a$ at a point $a$ is then naturally identified with the localization $k[x]_{(x-a)}$.
Now given an $O$-module $F$, each stalk $F_a$ will be a module over $k[x]_{(x-a)}$. But these module structures cannot be arbitrary: they have to be strongly related to each other! For instance, given any global section $s\in F(X)$, the actions of $x\in k[x]_{(x-a)}$ on the images of $s$ in each stalk $F_x$ have to be "the same": they all come from the action of $x\in k[x]=O(X)$ on $F(X)$ via the restriction maps. So for instance, if $F$ was a constant sheaf, then the $k[x]_{(x-a)}$-module structures on the stalks $F_x$ would have to all be the same, in that they actually glue together to make the constant value of $F$ a $k(x)$-module.