Is there a connection between the Indian word "Rama" (avatar of Vishnu) and the Semitic (Hebrew) word "Ram", meaning high, exalted, suprem)?

Possibly.

In the Vishnu sahasranama, Rama is the 394th name of Vishnu. "Rama" has two meanings: the supreme Brahman who is the eternally blissful spiritual Self in whom yogis delight or the One (i.e., Vishnu) who out of his own will assumed the enchanting form of Rama, the son of Dasaratha.

There is some speculation that the Hebrew character Abraham was also the Hindu Deity Ram.

Ram and Abraham were possibly the same person or clan. For
example, the syllable "Ab" or "Ap" means "father" in Kashmiri. The prototypical Jews could have called Ram "Ab-Ram" or "Father Ram."

It's also conceivable that the word "Brahm" evolved from "Ab-Ram" and not vice-versa. The Kashmiri word for "Divine Mercy," Raham, likewise derives from Ram. Ab-Raham = "Father of Divine Mercy."

The word "Abraham" is none other than a malpronunciation of the word Brahma. This can be clearly proven if one investigates the root meanings of both words. Abraham is said to be one of the oldest Semitic prophets. His name is supposed to be derived from the two Semitic words 'Ab' meaning 'Father' and 'Raam/Raham' meaning 'of the exalted.' In the book of Genesis, Abraham simply means 'Multitude.' The word Abraham is derived from the Sanskrit word Brahma.

The root of Brahma is 'Brah' which means - 'to grow or multiply in number.' In addition Lord Brahma, the Creator God of Hinduism is said to be the Father of all Men and Exalted of all the Gods, for it is from him that all beings were generated. Thus again we come to the meaning 'Exalted
Father.' This is a clear pointer that Abraham is none other than the heavenly father Brahma."

http://www.indiadivine.org/audarya/vedic-culture/189821-more-vedic-roots-abraham.html

https://en.wikipedia.org/wiki/Rama

It is certainly possible that the words in both languages share a common etymological background, the same origins and the same general ideas. Certainly there is a great deal of overlap between Semitic and Indo-European languages. (Though I'm betting that in this case, the Indian word is more derived from the Arabic or Aramaic, as Persian influances spread east with Zoroastrianism and Manicheanism and affected the Hindu culture. The word "ram" does not appear in the European tongues to which Indian is more closely related than to Hebrew.)