I am not aware of any spoken human languages that I am aware of that use only the silence between words, and ignore the spoken part. For spoken languages, this doesn't make much sense, but is not completely out of the realm of possibility. A language such as this would most likely not be the speaker's primary language. It would most likely have a specialized purpose.
There are some existing languages which are similar to your idea. There are several "drum languages" which are used primarily to communicate with people far away. The drums are used very much like Morse code over telegraph lines.
Drums in Communication
In Africa, New Guinea and the tropical America, people have used drum telegraphy to communicate with each other from far away for centuries. When European expeditions came into the jungles to explore the local forest, they were surprised to find that the message of their coming and their intention was carried through the woods a step in advance of their arrival. An African message can be transmitted at the speed of 100 miles in an hour.
There are also languages which use whistles, which has many similarities to the type of language you had in mind.
Whistled Language
Whistled languages differ according to whether the spoken language is tonal or not, with the whistling being either tone or articulation based (or both).
Tonal languages are often stripped of articulation, leaving only suprasegmental features such as duration and tone, and when whistled retain the spoken melodic line. Thus whistled tonal languages convey phonemic information solely through tone, length, and, to a lesser extent, stress, and most segmental phonemic distinctions of the spoken language are lost.
In non-tonal languages, more of the articulatory features of speech are retained, and the normally timbral variations imparted by the movements of the tongue and soft palate are transformed into pitch variations. 350
Certain consonants can be pronounced while whistling, so as to modify the whistled sound, much as consonants in spoken language modify the vowel sounds adjacent to them.
"All whistled languages share one basic characteristic: they function by varying the frequency of a simple wave-form as a function of time, generally with minimal dynamic variations, which is readily understandable since in most cases their only purpose is long-distance communication."
Most spoken languages make use of gaps in between words to convey meaning. These gaps are turned into punctuation marks when the language is converted to a written form. These gaps are well documented and the following example shows how they are used (emphasis mine).
Eats, Shoots & Leaves
A panda walks into a café. He orders a sandwich, eats it, then draws a gun and proceeds to fire it at the other patrons.
"Why?" asks the confused, surviving waiter amidst the carnage, as the panda makes towards the exit. The panda produces a badly punctuated wildlife manual and tosses it over his shoulder.
"Well, I'm a panda," he says. "Look it up."
The waiter turns to the relevant entry in the manual and, sure enough, finds an explanation. "Panda. Large black-and-white bear-like mammal, native to China. Eats, shoots and leaves." 312