Will there be capital gains/inheritance tax for the child inheriting a NY house valued at $2 million?

For the child - there are no immediate tax consequences, as there is no inheritance tax imposed either by the federal government or the state of New York.

When the property is sold or gifted at a later date the tax cost basis for the property is set to the fair market value of the property either as of the date of death of the decedent or alternatively 6 months later, at the executor's discretion, plus the costs of any capital improvements that are made to the property in the interim.

Of course it's possible that the estate may owe estate taxes, but that's a different topic.

Am I wrong to think that estate taxes are only instituted on value after the first $1 million?

Am I wrong to think that estate taxes are only instituted on value after the first $1 million?

Depends on the year that the decedant passed away. In 2008 it was $2 million, in 2009 it was $3.5 million, and then for 2010 the federal estate tax was abolished so the value was essentially unlimited. For 2011 the estate tax was reinstituted with an exemption of $5 million. But in any case - if estate tax is due it is paid out of the assets of the estate prior to distributing the remaining assets to the heirs, and is managed by the executor of the estate. The heirs do not pay it, and your original question was specifically about inheritance taxes to be paid by the heir.