As the question indicates, both Romeo and Juliet, as well as Lord Capulet and Paris, want hasty marriages. Romeo and Juliet are frantic to be married as quickly as possible, and Lord Capulet is very anxious for Paris to quickly marry Juliet, as Paris is himself.
Beyond the desire for a hasty marriage, however, Romeo is hasty in his response to the false news he hears that Juliet is dead. He immediately runs to the apothecary to buy poison to use to kill himself, and he then hurries back to Verona, despite being banned from the city on pain of death.
When he arrives at the Capulet burial vault, he never stops to question what is going on or whether Juliet is really dead. It never occurs to him to stop and see Friar Laurence and try to get some information from him or to find something strange in the sudden death of a young teenage girl. (People did die suddenly in that period, but nevertheless, Romeo acts with great speed.)
Romeo is too quick to kill himself on seeing what he believes is Juliet's corpse. If he had stopped for a few minutes to try to think things through, he would have given Juliet time to awaken from her feigned death, and the two could have been reunited. However, it was completely in character for Romeo to act emotionally and impulsively; it would not have been in his nature to stop and contemplate.
All this, along with the hasty street brawls the two feuding families jump into, reinforces the theme that too much haste leads to trouble and tragedy.