The answer is no, because although a mean-reverting process has necessarily to be stationary, it is not true the opposite, that is a stationary process has to be mean-reverting, as you stated in the question.
Look at this article for a formal definition of a mean-reverting process.
Now, think about one of the most famous mean-reverting process: the Ornstein–Uhlenbeck; the assumption underlying such process are the following:
The stationarity is a necessary assumption in order that such process is mean-reverting, but it is not true the opposite. The mean reverting process assumption is not a necessary condition such that a process is stationary.
For reference, look at the books I cited to have an idea of the application of such concepts in finance; you can find them online for free in pdf format. Since this is a quantitative finance site, I suppose the question indirectly refers to a pair-trading strategy, so, I think it is important to suggest you to read:
Chan, Ernest P. "Quantitative Trading." New Jersey (2008).
Particularly, read the paragraph about stationarity condition and cointegration (chp. 7). There, the author suggests:
. You can find a pair of stocks such that if you long one and short
the other, the market value of the pair is stationary. If this is the
case, then the two individual time series are said to be cointegrated.
They are so described because a linear combination of them is
integrated of order zero. 
Again, it is not true that a stationary process implies necessarily that the time series composing the process are cointegrated, but it is true the opposite ones; that is, if 2 time series are cointegrated, necessarily their linear combination (in the right proportions) is a stationary process.
Said that, in quantitative finance terms, it is not necessary that the 2 processes you are considering have to be stationary, but that it necessary their market value is stationary; in such case, the process resulting from the linear combination of those two time series is a good candidate for a pair-trading strategy.
Moreover, for a formal proof, look at:
Alexander, Carol. Market models: a guide to financial data analysis.
John Wiley & Sons, 2001.
He provides a formal proof of what I wrote down above and answers exactly to your question.