Question: Is, “Seeing happiness in their eyes really melt my heart”, correct?
Answer: The other answers already answered your query from grammatical point of view. From emotional point of view the expression, statement is true, correct, although for truly feeling this way one has to go through a fundamental self-transformation.
We are born with an inherently self-caring, self-justifying nature, which means that only our own happiness can truly melt our own heart. In order to be happy for the joy, happiness of others we would need to exit our inherently selfish, subjective spheres so we could sense reality through the desires of others, seeing the world through their eyes.
Then we become capable of accepting, even loving the others above the inherent self-love. Only then can we reach this special state when “their happiness melts our heart”.
Entering The Force-Field 