While romantic comedies certainly can make us laugh, and As You Like It certainly does contain some laughable moments, the definition of a romantic comedy actually goes beyond creating laughter.

One of the definitions is that a romantic comedy must contain some sort of tension or conflict, and the plot must move from establishing this conflict to resolving the conflict; the resolution will especially be a happy resolution. We certainly do see conflicts and resolutions in As You Like It. The central conflict is the fact that Duke Frederick has usurped his brother Duke Senior and driven him into exile in the Forest of Arden, along with Duke Senior's daughter Rosalind and Duke Frederick's own daughter Celia who decides to go into exile with Rosalind because Celia can't bear to be parted from Rosalind. This central conflict is resolved by the end of the play when Duke Frederick goes into the forest in pursuit of Duke Senior to kill him once and for all, but instead encounters a religious man who converts Duke Frederick to wanting to live a life in the forest and to restore the dukedom back to Duke Senior, as we see in the lines:

... And to the skirts of this wild wood he came;
Where meeting with an old religious man,
After some question with him, was converted
Both from his enterprise and from the world,
His crown now bequeathing to his banish'd brother. (V.iv.159-63)

The happy resolution of the play alone identifies the play as a romantic comedy.

Beyond happy resolutions, by definition, romantic comedies, especially Shakespearean romantic comedies, celebrate family and society. For that reason, the main action surrounding a romantic comedy surrounds the topic of love, such as Orlando's and Rosalind's love, plus any obstacles they overcome would be obstacles prohibiting a union through love. All of Shakespeare's romantic comedies end with a wedding, or at least plans for a wedding, which serves to celebrate family structure as well as social structure. The play As You Like It is certainly no different in that four weddings take place at the end of the play, showing us that the play's happy ending certainly is a celebration of family and society.

Hence, since the play offers a happy resolution to its central conflict, as well as other minor conflicts, plus ends in marriage, we know beyond a doubt that As You Like It fits the definition of a romantic comedy.

What makes Shakespeare's As You Like It a romantic comedy?

A romantic comedy can be defined as being one  

that presents the adventures of young lovers trying to overcome social, psychological, or interpersonal constraints to achieve a successful union.

As You Like It certainly fits this definition. Orlando and Rosalind are unknown to each other at the beginning of the play. Orlando's victory in the wrestling match, which starts as a reason for celebration in his eyes, becomes an obstacle when he and Rosalind become aware of the complications it presents to their futures.

DUKE FREDERICK.
What is thy name, young man?

ORLANDO.
Orlando, my liege; the youngest son of Sir Rowland de Bois.

DUKE FREDERICK.
I would thou hadst been son to some man else.
The world esteem'd thy father honourable,
But I did find him still mine enemy:
Thou shouldst have better pleas'd me with this deed
Hadst thou descended from another house.

...

ROSALIND.
He calls us back: my pride fell with my fortunes:
I'll ask him what he would.--Did you call, sir?--
Sir, you have wrestled well, and overthrown
More than your enemies.

And so the stage is set. Orlando and Rosalind, with Celia and Oliver supporting the intrigues of their friends. The complications and mistaken identities add to the comedic aspects of the play, while the romantic aspirations of all the characters drive the continuation of the action.

Shakespeare recognized the draw that romantic comedy had for the audiences that would be attending his plays, and As You Like It continues to be one of the most frequently producced of his works.