In the example you gave, it should be (with a few additional fixes):
I held the peculiar object gently in both hands as if it were made of fine glass. One nervous spasm and it could drop, shattering into a million shards, onto the hard wood floor. In his deep voice, my father exclaimed, “Happy birthday, son! Do you know what it is?”
Keeping my eyes fixed on the oddity in my hands, I nodded.
“Well, it's called a stethoscope. Can you say stethoscope?”
I repeated the sound.
“Very good,” my father replied. “Now, do you know what it's used for?”
I remembered that on my last birthday, my father gave me a microscope kit for children above the age of nine; it was my seventh birthday. Knowing that, and knowing that my father was a doctor, I knew the gift had something to do with medicine.
I gave him a puzzled look and he bent down on one knee, placed the eartips in my ears before pressing the other end up against his chest. My eyes widened as I heard the beating of his heart.
“Is that your heart, dad?” I asked in wonder.
“Yes it is son. You are listening to my heart.” My father handed me the flat end of the stethoscope, then he turned around and told me to press it up against his upper back. At first, I did not hear a thing but then, after my father took a deep breath, I let out a sigh. This time it was his breath bellowing in and out of his lungs.
Soon I was using it on myself, trying listen to my own body. My fathered chuckled with delight.
There are of course arguments for or against certain things, like exactly when the dialogue needs to be broken off into a new paragraph, but generally, particularly in the more flexible area of fiction or memoirs, if the person doing an action is the person who will speak, it can be in the same paragraph. So, for example, the line "Is that your heart, dad?" could go with the paragraph above. It also is dependent on how much you want to emphasize the line. In this case, it seemed like an important thing for the child to say, so it can absolutely have its own paragraph. The vital thing is that your readers do not become confused as to who is speaking.