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.