<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2021-04-12</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001823034</issuerCik>
        <issuerName>Hudson Executive Investment Corp. III</issuerName>
        <issuerTradingSymbol>HIII</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001847295</rptOwnerCik>
            <rptOwnerName>HEC Sponsor III LLC</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O HUDSON EXECUTIVE INVESTMENT CORP.</rptOwnerStreet1>
            <rptOwnerStreet2>III 570 LEXINGTON AVENUE, 35TH FLOOR</rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10022</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001298718</rptOwnerCik>
            <rptOwnerName>Bergeron Douglas</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O HUDSON EXECUTIVE INVESTMENT CORP.</rptOwnerStreet1>
            <rptOwnerStreet2>III, 570 LEXINGTON AVENUE, 35TH FLOOR</rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10022</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chief Executive Officer</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001495110</rptOwnerCik>
            <rptOwnerName>Braunstein Douglas L</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O HUDSON EXECUTIVE INVESTMENT CORP.</rptOwnerStreet1>
            <rptOwnerStreet2>III, 570 LEXINGTON AVENUE, 35TH FLOOR</rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10022</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>President, Chairman</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Class B Common Stock</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F1"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2021-04-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>J</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F2"/>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>93750</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F1"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F1"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Class A Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>93750</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>14900000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                    <footnoteId id="F3"/>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">The shares of Class B common stock are convertible for shares of the Issuer's Class A common stock as described under the heading &quot;Description of Securities&quot; in the Issuer's Registration Statement on Form S-1 (File No. 333-252744) (the &quot;Registration Statement&quot;) and have no expiration date.</footnote>
        <footnote id="F2">On April 12, 2021, HEC Sponsor III LLC forfeited at no cost 93,750 shares of Class B common stock, following the expiration of the over-allotment option granted to the underwriters of the Issuer's initial public offering (as more fully described in the Registration Statement), and the determination by such underwriters not to exercise in full such over-allotment option granted to them.</footnote>
        <footnote id="F3">HEC Sponsor III LLC is the record holder of the securities reported herein. Douglas G. Bergeron and Douglas L. Braunstein are among the managers of HEC Sponsor III LLC and share voting and investment discretion with respect to the securities held of record by HEC Sponsor III LLC. Mr. Braunstein and Mr. Bergeron disclaim any beneficial ownership of the securities held by HEC Sponsor III LLC other than to the extent of any pecuniary interest they may have therein, directly or indirectly.</footnote>
    </footnotes>

    <remarks>See Exhibit 99.1 - Joint Filer Information, which is incorporated herein by reference and describes in further detail the relationships of the Reporting Persons to the Issuer.</remarks>

    <ownerSignature>
        <signatureName>/s/ Douglas L. Braunstein, as Manager of HEC Sponsor III LLC</signatureName>
        <signatureDate>2021-04-14</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>/s/ Douglas G. Bergeron</signatureName>
        <signatureDate>2021-04-14</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>/s/ Douglas L. Braunstein</signatureName>
        <signatureDate>2021-04-14</signatureDate>
    </ownerSignature>
</ownershipDocument>
