displaying or showcasing the medal

A display case or cabinet, as in a store or museum.
A setting in which someone or something may be displayed, especially to advantage.

Above is the definition for showcase.

I find your question interesting. I hope you don't mind me pitching in my two cents.

If Andrew is holding the medal in his hand, I wouldn't use either. I don't think either is correct but am not certain.
The reason is according to the definition, it requires a glass cabinet or something and the medal must be placed in it for showcase to be acceptable. I believe display too.

Unless you meet the above criteria, I would use, showing off. Let's wait for a native speaker.

Hi,

Andrew is proudly displaying / showcasing the medal that he has won at the competition.

Which underlined word should I use?

'Displaying' is OK. In a context like this, it suggests 'showing some special thing in a special way'.

'Showcase' can be used as a verb. The media sometimes use it, although I don't hear ordinary people say it in ordinary conversation.

eg The museum is showcasing a Mayan death mask made of gold. This means it can be seen in their most important showcase.

eg The concert showcases Canada's most famous violinist. This means she is the most important performer in the show and everyone will get a chance to see her.

'Showcase' as a verb does not work in your example. A person does not usually showcase things.

Best wishes, Clive