Do I have to pay capital gains tax if proceeds were immediately used for another investment?

Buying another asset makes no difference, you still owe capital gains tax on the sale.

What cash should you use? If you didn't have other cash to pay the taxes you should have kept some of the cash from the sale to pay taxes and not reinvested it all.

You didn't mention what the asset was.

For non sheltered securities, their sale is a tax event whether it's a realized gain or loss regardless of what the holding period was or what you subsequently did with the money.

If the asset was say a property, the tax status would depend on the amount of the gain and whether it was your home or an investment property.

The IRS is not concerned with where you get the cash if taxes are due. You owe them and it's your responsibility to pay them.

It seems to me you are thinking of the 1031 Exchange which offers deferral of the gain if the proceeds are all invested in the next property. This is for (rental) real estate, not stocks. A similar rule applied to one's home, but that law is long gone, and instead there is an exemption under certain conditions.

I believe that, at least for stocks, you do not have to pay, or for that matter report CGT, if you are in the lowest tax brackets, perhaps up to 15%. I am quoting from memory, you should double check this.