The episode "A Solitary Place", has the Hulk saving a bunny from a bear trap. Then begins petting said bunny, gently and with care.
Likewise in "The Snare" when he places a bird back in its nest and strokes it gently.
A meta example is the episode King of the Beach, which seems to have been made primarily so that Bill Bixby and the showrunners could spend 40-odd minutes telling the audience what an awesome guy Lou Ferrigno is under the Hulk makeup and how he's far more than his muscles or his disability.
Every moment between Betty and Bruce is sweet. From her driving out in the rain to find him to cutting his hair and finding him clothes to willingly becoming a fugitive to help cure him. This troper's personal favorite moment is after Ross has captured them and is bringing them in, Betty slides her foot across to Bruce and smiles. He holds her hands and kisses them without saying anything. It's enough to make your heart melt.
During the university fight when General Ross orders the gunship to fire on the Hulk, with his daughter in the line of fire, the monster's first instinct rather than attack it (at first) is to shield Betty from the bullets.
Hulk knows exactly who Betty is and won't attack her.
The cave scene. After Hulk carries her in gently, Betty wakes up...and yells in fright. Hulk doesn't get angry - yep, the Hulkdoesn't get angry, and simply backs away into the rain in order to allow her to calm down. It doesn't take her long, and she approaches him. Then the thunder starts, and Hulk starts roaring at it and throwing rocks into the air, before signally for Betty to stay back - he thinks the thunder is a threat, and is trying to protect her from it. Then Betty leads him back inside, assuring him that everything's OK. The next morning, Betty wakes up lying on the now normal Bruce. If she had fallen asleep that way, she was willing to rest against him while he was the Hulk, and that says a lot.