Is it true that if marginal cost is constant, then average variable cost is also constant and equals marginal cost?
Your intuition is correct.
First, you're right that &quot;marginal cost only depends on variable cost&quot;, since
\begin{equation}
MC(q)=\frac{\mathrm dTC(q)}{\mathrm dq}=\frac{\mathrm d(FC+VC(q))}{\mathrm dq}=\frac{\mathrm dVC(q)}{\mathrm dq}. 
\end{equation}
Next, if marginal cost is some constant $k$, then variable cost must be $VC(q)=kq$, because we can integrate $MC$ to obtain $TC$, where the term that varies with $q$ is $kq$:
\begin{equation}
MC(q)=k 
\quad\Rightarrow\quad 
\int k\;\mathrm dq =\underbrace{kq}_{VC(q)} +\underbrace{K\vphantom{q}}_{FC} 
\end{equation}
where $K$ is a constant resulting from the integration.
Lastly,
\begin{equation}
AVC(q)=\frac{VC(q)}{q}=\frac{kq}{q}=k,
\end{equation}
which is a constant, and same as $MC$.