On transfer of shares, NRIs are required to pay capital gains tax in India.

2) The rate of tax depends on the period for which the shares were held by the NRI before transfer and whether the shares are listed or unlisted.

3) Listed shares held for more than 12 months are subject to long-term capital gains tax at the rate of 10% and those held for up to 12 months give rise to short-term capital gains, which are subject to tax at 15%.

4) Transfer/redemption of units of equity-oriented mutual funds held for a period exceeding 12 months are classified as long-term capital gains which are subject to tax at 10% and those held for up to 12 months are classified as short-term capital gains which are taxable at 15%.

5) In case of debt-oriented mutual funds, the holding period should be more than 36 months to qualify as long-term capital gains, which are taxable at 20%, whereas those held for up to 36 months are classified as short-term capital gains and are taxable at applicable slab rates.

Trading stocks in the U.S. market is not just for American citizens. While U.S. stocks and bonds are regulated by U.S. law, there are no explicit provisions prohibiting non-U.S. citizens from investing in the U.S. stock market and many investment firms cater to international clients who wish to purchase U.S. stocks.

The concepts and rules for determining the residential status income-tax laws and FEMA are quite different and it
would be possible to be a resident under one law and non-resident under the other.
2 For exemption of income tax in respect of NRE and FCNR deposits investor should be non-resident under FEMA.
3 The special tax rate concessions on income and long-term capital gains on specified assets, purchased in convertible foreign exchange are available to nonresidents under the Income-tax Act. 

Subject to the provisions of this Act, the total income of any previous year of a person who is a non-resident includes
all income from whatever source derived which
(a) is received or is deemed to be received in India in such year by or on behalf of such person; or

(b) accrues or arises or is deemed to accrue or arise to him in India during
such year.
Explanation I.-Income accruing or arising outside India shall not be deemed to be received in India within the meaning of this section by reason only of the fact that it is taken into account in a balance sheet prepared in India.
Explanation 2.-For the removal of doubts, it is hereby declared that income which has been included in the total income of a person on the basis that it has accrued or arisen or is deemed to have accrued or arisen to him shall not again be so included on the basis that it is received or deemed to be received by him in India.”
Thus, it is clear from the above that the incidence of tax depends upon a person’s Residential Status and also upon the place and time of accrual and receipt of income.

If there is taxation deducted in that country no need to pay in India

Capital gains tax arises on sale of shares 

merely because market value of shares have increased would not fasten any liability on you if you have not sold the shares 

Yes in both cases the tax will be applicable unless not charged in both countries. The rule of dual taxation will be applicable to you

ROR’s global income, including assets held abroad, is liable to tax in India.

An NRI or RNOR is not liable to income tax in India on his foreign-sourced incomes (unless received in India)

Capital gains earned in the US will be taxable in India if you qualify as resident and ordinarily resident (ROR) in India during the relevant financial year (FY).

The residency rules have changed with effect from FY21. You are a resident in India is if you satisfy any of the following conditions: (a) Physical presence in India during the relevant FY is 182 days or more; or (b) Physical presence during the relevant FY is 60 days or more and 365 days or more in the preceding four FYs. However, in respect of Indian citizens or persons of Indian origin (PIOs), who come on a visit to India, the second basic condition is modified

Since you are not a citizen of India, you may qualify as RNOR under either of the following three circumstances : (i) You are non-resident in India in nine of 10 FYs preceding the relevant FY; or (ii) Your physical presence in India is 729 days or less in the seven FYs preceding the relevant FY; or (iii) You are resident in India during the relevant FY due to 120 days residency rule.

An individual qualifying as ROR is taxed on his worldwide income in India and is required to report assets held outside India as also foreign incomes in the Indian tax return. An individual qualifying as a non-resident or RNOR is not liable to tax in India on his foreign-sourced incomes (unless received in India). As you are staying in India for more than a year, you may likely qualify as RNOR and, therefore, will be taxed on India-sourced income only. You will not be liable to tax in India on capital gains arising and received in the US.