Both are correct; however both are from a particular and somewhat refined English dialect (the kind you'd expect to read in a 19th century novel) so finding good illustrative examples is going to be difficult.
In this case "would" reflects its relationship with "will", as in intention or determination. The first sentence talks about something that has already happened and its consequence:
If he was so determined to bet on the races that he did so in spite of your objection, (then) it's his own fault that he lost his money.
The second sentence is a typical conditional that talks about some future hypothetical action:
If he is so determined to bet on the races that he will do so in spite of your objection, (then) it will be his own fault when he loses his money.
In either case, the "then" is optional.
I can't tell you which would definitely be correct on an "international exam", as many of these rely on so-called "rules" which are not always followed by actual English speakers and their many, many dialects. After all, something like:
It ain't no thing
would be marked incorrect on an exam, even though it's a fairly common expression.
If forced to pick between the two, the second sentence is a more standard example of a hypothetical and is probably correct. Probably.