The problem with the approach you have described is that those who are exposed to Injury $I$ are not mutually exclusive of those who are exposed to Condition $C$. So there are presumably individuals who have condition $C$ who are also Injured ($I$). But you also could have individuals who have condition $C$ who are not injured ($I$). As a result simply knowing $OR_1 > OR_2$ doesn't really tell you anything. Put another way, you can classify individuals into one of four categories:
So, if you look at the the odds ratio of individuals who have Disease $D$, when exposed to injury $I$, you are capturing $n_{11}$ as well as $n_{12}$ in the table above. When you then compare that odds-ratio to those who have disease $D$ when exposed to condition $C$, you are capturing those in $n_{11}$ and $n_{21}$ in the table above. As you can see, both odds-ratios will involve the same individuals from $n_{11}$. The problem here essentially boils down to contamination of the comparison. In other words, you are changing multiple variables at the same time making it impossible to determine the association between an exposure and the disease. To see this more easily, it's helpful to write out the possible categories of disease and exposures into which a person can fall. The tables below will help with this.
Recall that the odds ratio $OR$ can be written as:
\begin{eqnarray*}
OR & = & \frac{{\color{blue}{\color{blue}P(Disease|Exposure)}}}{1-P(Disease|Exposure)}/\frac{P(No\,Disease|Exposure)}{P(No\,Disease|Exposure)}
\end{eqnarray*}
It's also helpful to focus on the top Numerator $NOR$ in each of the odds ratios ($OR_1$ and $OR_2$) which, more or less, are compared in your analysis. This is the term in blue font above. The can be written as:
\begin{eqnarray*}
NOR_{1} & = & P(Disease|Injury)\\
& = & P(Disease|Injury\,and\,Condition)+P(Disease|Injury\,and\,No\,Condition)
\end{eqnarray*}
\
\begin{eqnarray*}
NOR_{2} & = & P(Disease|Condition)\\
& = & P(Disease|Injury\,and\,Condition)+P(Disease|No\,Injury\,and\,Condition)
\end{eqnarray*}
The comparison between $OR_1$ and $OR_2$ can largely be boiled down to comparing these two terms, $NOR_1$ and $NOR_2$. When you do so, you see the that $P(Disease|Injury\,and\,Condition)$ terms cancel each other out, since it appears in both $NOR_1$ and $NOR_2$, and we are left comparing:
$P(Disease|Injury\,and\,No\,Condition)$ from $NOR_1$ with $P(Disease|No\,Injury\,and\,Condition)$ from $NOR_2$.
But you see that this is a comparison of those who have the disease given they were injured and not exposed to condition $C$ to those who were not injured and were exposed to condition $C$. So both the injury exposure and condition exposure are changing. This makes it impossible to determine if it is the injury exposure or the condition exposure that is associated with the changes in disease state.
What is likely going to be a better analysis for you is to perform a logistic regression where you regress the log of the odds of disease (the logit) onto condition, injury and their interaction (see here for a seminar on this). This will allow you to disentangle and determine which is the more important factor associated with disease state.
