What is the speaking in tongues (as referred to in Acts 2:4)?
Key Verse:  And how hear we every man in our own tongue, wherein we were born.  [Acts 2:8]

Is speaking in tongues necessary to prove the Holy Spirit is dwelling in an individual?

There is a teaching today which indicates that if one is able to "speak in an unknown tongue" that it is proof the Holy Spirit is within the individual.  Can we also believe that if one does not "speak in an unknown tongue" that God's spirit is not in them?

While people strain to prove God is working with them, we are clearly told what the fruit of the Spirit is in Galatians.
But the fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith, Meekness, temperance: against such there is no law. [Galatians 5:22-23]
The practice of "speaking in an unknown tongue" is correctly called glossolalia. Webster's Dictionary gives the following definition: "an ecstatic of apparently eustatic utterance of unintelligible speechlike sounds, viewed by some as a maninfestation of deep religious experiences".  This glossolalia has been passed off as a holy language, so that people doing this try to show others that they have a gift.

In I Corinthians 14 we read the following:
He that speaketh in an unknown tongue edifieth himself; but he that prophesieth edifieth the church.  I would that ye all spake with tongues, but rather that ye prophesied: for greater is he that prophesieth than he that speaketh with tongues, except he interpret, that the church may receive edifying.  Now, brethren, if I come unto you speaking with tongues, what shall I profit you, except I shall speak to you either by revelation, or by knowledge, or by prophesying, or by doctrine?  And even things without life giving sound, whether pipe or harp, except they give a distinction in the sounds, how shall it be known what is piped or harped?  For if the trumpet give an uncertain sound, who shall prepare himself to the battle?  So likewise ye, except ye utter by the tongue words easy to be understood, how shall it be known what is spoken? for ye shall speak into the air.  There are, it may be, so many kinds of voices in the world, and none of them is without signification.  Therefore if I know not the meaning of the voice, I shall be unto him that speaketh a barbarian, and he that speaketh shall be a barbarian unto me.  Even so ye, forasmuch as ye are zealous of spiritual gifts, seek that ye may excel to the edifying of the church. Wherefore let him that speaketh in an unknown tongue pray that he may interpret.  For if I pray in an unknown tongue, my spirit prayeth, but my understanding is unfruitful.  What is it then? I will pray with the spirit, and I will pray with the understanding also: I will sing with the spirit, and I will sing with the understanding also.  [1 Corinthians 14:4-15]
The Apostle Paul writes about the unknown tongue. Two things must be taken into consideration when studying this chapter. First, the word unknown is in italics in the authorized King James version, which indicates it was added.  Secondly, the word tongue should be translated language.  Paul is explaining that to speak in a language (different from those present) edifies only the speaker while prophesying (teaching in their language) edifies the church or the hearers.

When the apostles were anointed with the Holy Spirit on the day of Pentecost, they spoke different languages than their own, but the languages were understood by the hearers.
And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance.  And there were dwelling at Jerusalem Jews, devout men, out of every nation under heaven.  Now when this was noised abroad, the multitude came together, and were confounded, because that every man heard them speak in his own language.  And they were all amazed and marvelled, saying one to another, Behold, are not all these which speak Galilaeans?  And how hear we every man in our own tongue, wherein we were born.  [Acts 2:4-8]
Verse eight says "And how hear we every man in our own tongue (language), wherein we were born".  In other words, the apostles were speaking in a known language (understood without an interpreter) that the men had spoken since their birth.

Lastly, we are told this gift is for believers.  It is a necessary gift.  It was a gift given to the believer for a sign to the unbeliever.
Wherefore tongues are for a sign, not to them that believe, but to them that believe not: but prophesying serveth not for them that believe not, but for them which believe  [Corinthians 14:22]
The disciples were prepared for missionary work to take the gospel worldwide.  To prepare these Galilaeans, (Acts 2:7) they were anointed so that they could speak where ever they went in a recognizable, intelligent tongue.

It is well known that some groups teach that the speaking in unknown tongues is a sign that the individual has the Holy Spirit.  Many individuals try and help others to get started speaking in the unknown tongue by practice and mouth formations.  This might be called priming the pump as it were.  Friends, God does not need help.  Let us recognize this teaching for what it is - glossolalia, unintelligible speechlike sounds.  While speaking in tongues (languages) is a gift to help an individual, who does not speak the language, understand God's Word, the unintelligable jabberring is not edifying and is not from God.